refs.bib

# A Survey of Scam Exposure, Victimization, Types, Vectors, and Reporting in
12 Countries

Mo Houtti  Abhishek Roy  Venkata Narsi Reddy Gangula  Ashley Marie Walker

###### Abstract

Scams are a widespread issue with severe consequences for both victims and
perpetrators, but existing data collection is fragmented, precluding global
and comparative local understanding. The present study addresses this gap
through a nationally representative survey (n = 8,369) on scam exposure,
victimization, types, vectors, and reporting in 12 countries: Belgium, Egypt,
France, Hungary, Indonesia, Mexico, Romania, Slovakia, South Africa, South
Korea, Sweden, and the United Kingdom. We analyze 6 survey questions to build
a detailed quantitative picture of the scams landscape in each country, and
compare across countries to identify global patterns. We find, first, that
residents of less affluent countries suffer financial loss from scams more
often. Second, we find that the internet plays a key role in scams across the
globe, and that GNI per-capita is strongly associated with specific scam types
and contact vectors. Third, we find widespread under-reporting, with residents
of less affluent countries being less likely to know how to report a scam. Our
findings contribute valuable insights for researchers, practitioners, and
policymakers in the online fraud and scam prevention space.

##  1 Introduction

Scams affect many people. A 2017 FTC survey found that almost 16% of US
consumers had been victims of fraud in the past year Anderson2019-ln. This
problem extends worldwide; a 2023 global survey by the Global Anti-Scam
Alliance (GASA) found that 78% of respondents had experienced a scam in the
past year Abraham_undated-uy. The material consequences of scams and fraud can
be severe, extending far beyond moderate financial losses for consumers. There
are many documented cases of individuals losing their life savings to scams,
leading to severe material and emotional harm Coakley2024-cm, Khandro2024-uj,
Farivar2022-jj, Kilmer2023-zn. In some cases, businesses have been destroyed
by losing immense sums of money in a scam Albright2023-wt. But even these
facts focus on only one set of victims. The United Nations estimates that
hundreds of thousands of human trafficking victims are forced by criminal
organizations to carry out scams, often after being lured to a foreign country
with false promises of a job opportunity
United_Nations_Human_Rights_Office_of_the_High_Commissioner2023-mp. In
essence, the prevalence of scams has created lucrative opportunities for
organized crime, with severe consequences for both victims and perpetrators.

The scope of this issue will likely become more important as internet access
and the availability of digital payments continue to increase Fico_undated-ir.
The growing prevalence and widespread access to generative AI tools, such as
voice cloning and high fidelity image manipulation technology, could also
exacerbate the scam problem even further. For example, in a recent incident,
perpetrators used AI-generated voice and images to impersonate a company’s
CFO, successfully tricking a Hong Kong finance worker into wiring $25.6
million to fraudulent bank accounts Ianzito2024-gv. Proliferation of such
tools could enable scams to be a lot more believable and harder to discern for
users Bethea2024-dn. Unless curbed, this could increase the likelihood of
victimization, and potentially reduce trust in online interactions and
transactions as a whole FCC2024-dj.

To help better understand and combat this problem, governments around the
world regularly collect data on scams in their local jurisdictions. The
Federal Trade Commission (FTC), for example, conducts surveys and publishes
detailed reports Anderson2019-ln, noauthor_2023-dp to map the landscape of
scams in the United States. The European Commission assumes similar
responsibilities in the EU European_Anti-Fraud_Office_undated-qa, and
analogous data collection efforts happen in places like the United Kingdom
Jones2022-no, Australia abs_personal_2024, and South Korea Jun-hee2023-jo.

But despite these efforts, we do not have a good quantitative understanding of
scams at a global level. Because data collection typically happens within
jurisdictions, methodological differences make it impossible to compare
findings across borders. While some global reports do exist, they either
compile data from existing sources or use survey methodologies that do not
achieve population-representative samples (e.g., Abraham_undated-uy,
Lewis2020-kt). Some industry firms (e.g., NortonLifeLock2022-wi) have
conducted representative global surveys, but these have tended to be more
high-level and are therefore insufficiently granular to let us draw precise
and practical conclusions about the global scams landscape. This in turn
hinders policymakers, practitioners, and researchers from identifying and
targeting scams in contextually appropriate ways.

To address this gap, we conducted a nationally representative survey (n =
8,369) in 12 countries—Belgium, Egypt, France, Hungary, Indonesia, Mexico,
Romania, Slovakia, South Africa, South Korea, Sweden, and the United Kingdom.
The survey asked people about their experiences with scams, including the type
of scam they had most recently experienced, the vector through which they
experienced the scam, whether they had sent the scammer money, and
whether/where they reported the scam. Importantly, we ask these questions
through a global and nationally representative survey, making it possible to
draw true comparisons and contrasts between countries. This provides a couple
key advantages. First, it lets us contextualize local findings. If we find
that 25% of South Africans report having lost money from a scam in the past
year, is that a lot? According to our survey, the answer in this case is
yes—but this argument would be much more difficult to make in the absence of
global data. Second, while not a perfect substitute for comprehensive global
data, analyzing scam data across a broad spectrum of countries can help
resource-constrained governments prioritize their efforts. If consistent scam
types emerge across diverse economies, or if similar economies suffer from
specific scam types, they can be valuable signals to guide local data
collection and enforcement efforts. Concretely, the survey let us answer the
following research questions in multiple countries:

  * •

RQ1: How common are scam exposure and victimization?

  * •

RQ2: How prevalent are specific scam types?

  * •

RQ3: To what extent are scams technology-mediated?

  * •

RQ4: How comprehensive are reporting-based data sources about scam
victimization?

We present several useful findings, from which we contribute concrete
takeaways for researchers, practitioners, and policymakers in the online
safety and fraud prevention spaces. First, we find that residents of less
affluent countries suffer financial loss from scams more often. Second, we
find that the internet plays a key role in scams across all surveyed
countries, but that GNI per-capita is strongly correlated with specific scam
types and contact methods from scammers. For example, money-making scams are
more common in less affluent countries. Third, we find widespread under-
reporting of scams. On overage, half of those exposed to a scam in a given
country do not report it at all—and residents of less affluent countries are
more likely to not know how to report a scam.

##  2 Background

###  2.1 Scam Exposure and Victimization

Scams are a growing issue, with losses from reported scams increasing at a
rapid rate. Industry surveys and reports compiled by law enforcement are the
primary sources of scam prevalence data. The FBI’s Internet Crime Report 2023
showed that losses grew at upwards of 20% year over year in the last three
years fbi_ic3_2023. The Global Anti-Scam Alliance (GASA)’s State of Scams 2023
report estimated that losses from scams topped USD 1 trillion in 2023 and
accounted for about 1% of global GDP Abraham_undated-uy. While estimates for
exposure to scams vary widely across surveys and countries, it is generally
accepted that the exposure to scams is increasing year over year. GASA’s 2023
report stated that 78% of participants experienced at least one scam in the
past 12 months.

The GASA report estimated that developing countries like Kenya, Vietnam,
Thailand and Brazil lost more than 3% of their GDP to scams Abraham_undated-
uy. A 2020 survey commissioned by the European Commission (EC) found that the
exposure to fraud is generally higher in “connected countries” identified as
countries with a relatively high proportion of individuals making internet
purchases ec_scams_survey. In the context of online scams, increased internet
activity (e.g., online purchases) expands the pool of potential targets and
the opportunities for motivated offenders kigerl2012routine, felson1980RAT,
miro2014routine. Prior research indicates that a variety of psychological
(e.g. overconfidence, optimism bias), socio-demographic (e.g. age, education)
and situational factors (e.g. emotional distress, financial strain) play a
role in susceptibility to scams Whitty2019-hv,Modic2013-wn,Button2014-ip.

In this study, we looked at how the exposure and victimization to scams vary
across geographies.

###  2.2 Scam Types and Technology’s Role in Scams

Scams are heterogeneous and evolve based on a variety of factors such as
emerging technologies (e.g., cryptocurrency), changing communication channels
(e.g., social media), and global events (e.g., the COVID-19 pandemic, the
Russia-Ukraine war) xu2022differentiating. Scammers adapt their methods based
on the individual circumstances of their targets, ranging from preying on
financial vulnerabilities through lottery scams to exploiting loneliness
through romance scams. While attempts have been made by multiple government
agencies and researchers, there is still no universally accepted
classification of scams. DeLiema et al. classify consumer fraud into four
categories: opportunity-based scams, threat-based scams, consumer purchase
scams, and phishing scams DeLiema2023-yo. In contrast, FINRA’s Financial Fraud
Research Center has developed a framework for a taxonomy of fraud modeled
after international crime classification systems beals2015framework. In this
study, we used a scam classification based on the role scammers play and the
service they purportedly offer to understand the prevalence of such scams.

As stated earlier, scammers take advantage of technological advances to
effectively reach and scam their targets. Scammers use a variety of
communication channels such as phone calls, text messages, emails, social
media, and mobile apps to establish contact with their targets. Scammers also
leverage advances in payment methods with the emergence of irreversible
payment methods such as real-time payments, cryptocurrency, digital wallets,
etc. Notably, scams utilizing cryptocurrency have skyrocketed, with the FTC
reporting that cryptocurrency scams make up more than 85% of losses due to
investment scams noauthor_2023-dp. In this study, we looked at how scammers
use technology to reach their targets and receive payments across countries of
interest.

###  2.3 Limitations of Reporting-based Data Sources

Despite scams leading to devastating financial losses and causing enormous
psychological distress to victims munton2023scams, they are highly under-
reported to authorities with the FTC estimating that only 10-12% of scams are
actually reported to them noauthor_2023-dp. Deliema2019-bw show that scams are
under-acknowledged and under-reported even in survey research, likely due to
social desirability bias or refusal to acknowledge victimization. In the
limited studies that surveyed known victim pools, only about half of known
fraud victims admitted to being defrauded
button2009fraud,finra_sr_fraud_risk_survey. This shows that survey-based
estimates are better than reporting-based estimates to assess scam prevalence
and victimization.

While reports like those from the FBI and industry players provide valuable
information, they suffer from methodological disadvantages that hinder their
representativeness. Reports based on complaint data do not provide a complete
picture, as scams are severely under-reported to government agencies. Some
surveys by industry players are global but often focus on a narrow aspect of
the problem such as real-time payments based scams or text message scams, or
have representativeness issues. Through this survey, we looked at scam
exposure, victimization and reporting attitudes across regions using a
representative sample of participants in order to identify regional
disparities and inform targeted prevention and intervention strategies.

##  3 Methods

###  3.1 Survey Overview

The questions analyzed in this paper were part of a larger survey covering a
variety of digital safety issues. The survey was deployed through Morning
Consult (MC)—a leading survey research firm—using a mixed-panels approach,
which leverages the strengths of different data collection methods to create a
more comprehensive and representative picture of public opinion. MC uses
roughly 55 survey panel providers to conduct interviews across numerous
countries. This panel network provides access to tens of millions of survey
respondents via recruitment from thousands of websites, mobile apps, social
networks, email lists, and publishers. They use a wide mix of panel providers
with different recruitment methods to diversify their respondent pools and
ensure maximal access to different respondent groups. Recruited participants
completed the survey online, using mobile or desktop devices.

12 countries were surveyed: Belgium, Egypt, France, Hungary, Indonesia,
Mexico, Romania, Slovakia, South Africa, South Korea, Sweden, and the United
Kingdom. Questions were translated by teams with native speaker(s) in the
target languages through a process consisting of translation, editing,
proofreading, and quality assurance. Demographic factors used for stratified
sampling and weighting varied by country to account for contextual factors
such as the availability of reliable census data. Weights and sample targets
were based on general population proportions for adults (18 and older) in
Belgium, Egypt, France, Hungary, Indonesia, Romania, Slovakia, South Korea,
Sweden, and the UK. Mexico and South Africa have internet access penetration
below 80%, and folks without access to the internet are more likely to fall
into lower education demographics. Therefore, to avoid over-representing
adults with lower education, weights and sample targets for those two
countries were based on internet population proportions for adults (18 and
older). Responses were weighted using standard raking procedures
Battaglia2009-ad. Table 1 reports the demographic factors used to derive
sample targets and weights in each country.

Table 1: Demographic factors used for stratified sampling and weighting in each country. Country | Sample size | Sample targets based on | Weighting dimensions
---|---|---|---
Belgium | 504 | Age, gender | Age, gender, education, region
Egypt | 740 | Age, gender | Age, gender, education, region
France | 738 | Age, gender | Age, gender, education, region
Hungary | 527 | Age, gender | Age, gender
Indonesia | 752 | Age, gender | Age, gender, education, region
Mexico | 778 | Age, gender, education | Age, gender, education, region
Romania | 761 | Age, gender | Age, gender, education, region
Slovakia | 724 | Age, gender | Age, gender
South Africa | 773 | Age, gender, education | Age, gender
South Korea | 782 | Age, gender | Age, gender
Sweden | 533 | Age, gender | Age, gender, education, region
United Kingdom | 757 | Age, gender, education | Age, gender, education, region

###  3.2 Survey Questions

We used the FTC’s online fraud reporting tool
Federal_Trade_Commission_undated-hm as a guide when formulating our survey
questions. This let us ensure that the questions captured the kind of
information about scams that is useful to government institutions and that the
multiple-choice options adequately covered common scam experiences. To account
for the likelihood of multiple scam experiences, respondents were instructed
to answer questions based on their most recent scam experience (if any) in the
past year. We reasoned that respondents might have trouble recalling details
of earlier scam experiences, making their reports less reliable. Obtaining a
cross-section of most recent scam experiences would also closely approximate
the breakdown of scam experiences overall while letting us simplify the survey
for respondents.

While “scams” and “fraud” technically have different definitions, the two
terms are often used interchangeably to refer to attempts to trick someone for
monetary gain. Indeed, the aforementioned FTC reporting tool
Federal_Trade_Commission_undated-hm does not distinguish between the two.
Rather than providing a narrow definition that might unintentionally exclude
instances commonly considered to be scams, we decided to mirror the FTC and
rely on respondents’ colloquial understanding of the term.

Recall that we articulated four research questions. We state each research
question before its corresponding survey questions.

RQ1: How common are scam exposure and victimization?

We analyzed responses to Q1 to understand the frequency with which internet
users are exposed to and materially harmed by scams:

Q1: In the past year, have you been the target of a scam?

  * •

Yes, I have been the target of a scam, but I did not lose money.

  * •

Yes, I have been the target of a scam and I did lose money.

  * •

I have not been targeted by any scams.

  * •

I do not know if I’ve been targeted by a scam.

RQ2: How prevalent are specific scam types?

We analyzed Q2 to understand the types of scams internet users experience:

Q2: What kind of scam were you targeted by? Choose the one that most
accurately describes your experience.

  * •

An impersonator (e.g., fake government, business, love interest, grandchild)

  * •

Job, investment, money-making opportunity, franchise

  * •

Phone, internet, TV service

  * •

Health (e.g., weight loss, eye care, treatment)

  * •

Online shopping (e.g. fake stores/pages)

  * •

Sweepstakes, prize, lottery

  * •

Auto sale, repair

  * •

Credit, debt, loan (e.g., debt collection, credit report, student loan debt
relief)

  * •

Something else

RQ3: To what extent are scams technology-mediated?

Responses to Q3 and Q4 let us understand to what extent users’ scam
experiences are technology-mediated. Q4 was only given to respondents who
indicated having directly sent the scammer money in a separate question.

Q3: How did it start (e.g., how did they first contact you, where did you see
an ad)?

  * •

Phone call

  * •

Social media (e.g., Facebook/Meta, Instagram)

  * •

Online ad or Pop-up

  * •

Website or App (e.g., Craigslist, OfferUp, Marketplace)

  * •

Messaging app (e.g., Google Hangouts, WhatsApp)

  * •

Email

  * •

Text

  * •

Mail

  * •

In-person

  * •

TV or Radio

  * •

Print media (e.g., newspaper, magazine)

  * •

Other (please specify)

Q4: How did you pay or send the money?

  * •

Scammer transferred from my account without my consent

  * •

Paid through a payment or banking app on phone

  * •

Paid via payment or banking website on computer

  * •

Mailed cash, cheque, or money order

  * •

Bought gift cards & shared the details with the scammer

  * •

Other (please specify)

RQ4: How comprehensive are reporting-based data sources about scam
victimization?

Finally Q5 and Q6 let us contextualize the current research information
landscape on scams, which often relies heavily on reporting-based data.

Q5: Did you report the scam to anyone?

  * •

Yes

  * •

No

  * •

I wanted to, but didn’t know how

Those who did report the scam were asked:

Q6: Who did you report the scam to?

  * •

To a financial authority (e.g., bank)

  * •

To the service provider (e.g. Cisco, IBM, McAfee)

  * •

To the authorities (e.g., the police)

  * •

To my email provider

  * •

To the online security provider

  * •

To my network provider

  * •

To my family (who take action on my behalf)

  * •

To my work/place of education

###  3.3 Analysis

For each survey question, we report the weighted percentage of respondents who
selected each multiple-choice option in a heatmap. Countries are ordered along
the x-axis by ascending GNI per-capita in 2021 (the most recent year for which
complete data was available). Where the heatmaps suggest possible GNI-based
patterns among the most frequently selected or most pertinent options, we
compute Spearman’s rank-order correlations to verify and quantify the
associations.

##  4 Results

RQ1: How common are scam exposure and victimization?

###  4.1 Users in Less Affluent Countries Are at Greater Risk

On average, 15% of a country’s internet users lost money from scams in the
past year (Figure 1). Our results also reveal a troubling pattern: internet
users in less affluent countries are at greater risk of falling victim to
scams. South Africa, for example, had a GNI per-capita of $12,948 and the
highest scam victimization rate at 25%. South Korea, with a GNI per-capita of
$44,501, had less than a third of South Africa’s victimization rate (7%). We
found a strong inverse correlation between GNI per-capita and scam
victimization rate ($\rho=-0.73$, $p=0.007$). This is especially concerning
because lower incomes in countries with lower GNI per-capita make it more
difficult for victims to recover from financial loss.

###  4.2 Scam Exposure Does Not Reduce Victimization Rate

Some countries with low rates of scam exposure nevertheless have a high rate
at which those who do experience scams actually suffer losses (e.g., Egypt,
Mexico, and Sweden). Users in these countries may simultaneously be well
protected from exposure to scams, but not resilient against scams themselves.
Indeed, one could reasonably hypothesize that less frequent scam exposure
might make users less alert and therefore less resilient overall.

However, we found no correlation in either direction between the scam loss
rate—the top row in Figure 1—and scam exposure rate—the sum of the following
two rows ($\rho=0.04$, $p=0.90$). This suggests a more complicated
relationship between scam exposure and avoidance of financial loss from scams,
with additional factors possibly at play. For example, countries where users
are frequently exposed to scams may become more resilient in absolute terms,
but be targeted using more sophisticated techniques in response. This kind of
tailoring would be consistent with the results we cover next, demonstrating
some differences in scam types and communication vectors based on countries’
affluence.

Figure 1: Weighted within-country proportions of responses to Q1: In the past
year, have you been the target of a scam? Darker colors gradually indicate
higher values. The top row describes the percent who lost money among those
who experienced a scam; it is computed from the two rows immediately below it.

RQ2: How prevalent are specific scam types?

###  4.3 Top Scam Types Are Fairly Consistent Across Countries

The two most common scam types were online shopping and money-making scams
(Figure 2). There were only two countries where neither of these was the most-
selected option: Indonesia with sweepstakes/lottery scams (30%), and South
Korea with impersonation scams (28%). In both cases, however, money-making and
online shopping scams were still in second and third place, respectively.

Despite this consistency, we did see differences based on countries’
economies. Money-making scams were more common in less affluent countries; GNI
per-capita and the prevalence of money-making scams had a very strong inverse
correlation ($\rho=-0.90$, $p<0.001$). The prevalence of online shopping scams
was not significantly correlated with GNI per-capita ($\rho=0.44$, $p=0.15$).

Figure 2: Weighted within-country proportions of responses to Q2: What kind of
scam were you targeted by? Choose the one that most accurately describes your
experience. Darker colors gradually indicate higher values.

RQ3: To what extent are scams technology-mediated?

###  4.4 From First Contact to Payment, the Internet Plays a Key Role

Consistent with FTC reports in the US noauthor_2023-dp, the most common method
of first contact across countries (Figure 3) was social media (24%), followed
by email (16%) and phone (14%). On average, 68% of those exposed to a scam in
a country had first contact with a scammer through an internet-based
technology (social media, email, messaging app, website or app, or online ad
or pop up). Social media and messaging apps were more common as initial
contact methods in less affluent countries, while email-based scams were more
prominent in more affluent countries. GNI per-capita had a strong inverse
correlation with the prevalence of social media ($\rho=-0.71$, $p=0.009$) and
messaging apps ($\rho=-0.65$, $p=0.02$) as first contact methods, and a strong
positive correlation with email ($\rho=0.67$, $p=0.02$). Interestingly, South
Korea was a major outlier in our data; phone calls and texts were the most
common contact methods there, with internet-based methods making up only 30%.

Figure 3: Weighted within-country proportions of responses to Q3: How did it
start (e.g., how did they first contact you, where did you see an ad)? Darker
colors gradually indicate higher values.

Mobile apps were a key vector for payments to scammers. Mobile-based payments
were almost twice as common as computer-based ones (Figure 4). This pattern
held across most countries—with France and Belgium being the only exceptions.
This may simply reflect online banking habits across countries, but
nonetheless provides good evidence that scam prevention should include a focus
on mobile payments, and further substantiates reports calling out the risks of
the increasing global adoption of real-time payments (RTP) Fico_undated-ir.
The prevalence of mobile-based payments to a scammer had a strong inverse
correlation with GNI per-capita ($\rho=-0.79$, $p=0.002$), suggesting the
adverse effects of RTP adoption may be more prominent in less affluent
countries.

Figure 4: Weighted within-country proportions of responses to Q4: How did you
pay or send the money? Darker colors gradually indicate higher values.

RQ4: How comprehensive are reporting-based data sources about scam
victimization?

###  4.5 Many Scams Go Unreported

Our findings substantiate the widespread understanding that scams are under-
reported, and provide quantitative evidence of the extent to which this is
true globally. On average, over a third (37%) of a country’s scam victims did
not report the scam to anyone (Figure 5). In Slovakia and South Korea, that
number was closer to half (50% and 54%, respectively). Users in less affluent
countries more often wanted to report scams but did not know how
($\rho=-0.82$, $p=0.001$). This suggests that inferences based on victim
reports are likely to suffer more under-reporting bias in resource-constrained
countries, where survey data is also less likely to exist.

Figure 5: Weighted within-country proportions of responses to Q5: Did you
report the scam to anyone? Darker colors gradually indicate higher values.

Reporting to government authorities constituted an even smaller fraction of
these totals (Figure 6). Of those who reported scams, less than a third (27%)
reported them to the authorities. This varied widely by country—e.g., 57% of
South Koreans who reported scams reported them to authorities versus only 16%
of South Africans. GNI per-capita did not correlate significantly with
reporting to either authorities ($\rho=0.47$, $p=0.12$) or financial
institutions ($\rho=0.19$, $p=0.56$), though the proportions may reflect the
degree to which victims in each country believe a particular institution may
be able to take action for them (e.g., retrieve their lost money).

Figure 6: Weighted within-country proportions of responses to Q6: Who did you
report the scam to? Darker colors gradually indicate higher values.

##  5 Discussion

Scams are a widespread and growing problem affecting people worldwide. The
recent cost-of-living crisis Economist_Intelligence_Unit2023-nl has
exacerbated the financial impact of scams, leaving victims even more
vulnerable. Greater rates of financial loss from scam victimization coupled
with more difficulty recovering from financial loss (due to low income) in
less affluent economies makes this issue worth even more attention.
Consequently, governments, central banks, industry and civil society around
the world have responded with various tools in their arsenal, from public
awareness campaigns to regulationsStainsby2023-ue.

This lack of standardized data hinders our ability to understand and address
scam victimization effectively. Without a global survey on scam prevalence and
victimization, it is hard to accurately assess and describe the true size of
the issue, develop effective solutions or bring appropriate resources to bear.
Our work tries to fill in some of the gaps in terms of understanding who is
most at risk, and to what types of scams, so regulators and industry can
better target their interventions. Furthermore, as our numbers on contact
method and payment method show, the scams landscapes in low GNI per-capita
countries often look different, so minority-world-focused solutions are likely
not going to be as effective in the majority world. This survey aims to fill
that gap in literature.

In this section, we highlight some of the key drivers of scam victimization
across countries as demonstrated by our findings, and define a multi-level
structural approach to fight back against this issue.

###  5.1 Drivers of Scam Victimization Across Countries

While the specific tactics and methods used by scammers vary across countries,
there are several common tailwinds and themes contributing to the widespread
vulnerability and victimization to scams.

####  5.1.1 Widespread and fast adoption of digital payment systems globally
post-COVID appears to be a key driver in increased scam prevalence.

As we show in Figure 3, digital payment solutions are involved in a large
fraction of the scams incidents across all the countries we surveyed. This
observation mirrors the growth of the total number of non-cash transactions
happening globally, which is estimated to accelerate to $2.3 trillion by 2027,
growing at a rate of 15% annually Capgemini2023-rc.

Frauds and scams are not, however, unique to digital payment systems. While
digital payment systems are fast, convenient and thereby have a large volume,
they often offer limited recourse for victims as compared to modes of payment
like credit cards. While with the adoption of such payment systems we have
seen a massive growth in economic activity, this has also increased the total
number of scam activities over time. For example, UK’s Payment Systems
Regulator (PSR) found just 0.1% of fast payments in 2021 were fraudulent—well
above the global average of 0.03% for card transactions World_Bank2023-yq.
While this trend of growth in adoption of digital payments has implications
for the total volume of scams occurring globally, it is particularly pertinent
in Asia Pacific where non-cash transactions seem to be growing at an estimated
rate of 19.8% annually Capgemini2023-rc. The type of scams that are prevalent
vary country to country, but it appears that focusing research and enforcement
efforts on scams facilitated through digital payment services is a productive
intervention space, especially for those countries.

####  5.1.2 Scams follow where users go online, and locale specific factors
are highly salient in understanding specific scam victimization.

As we show in Figure 2, the types of scams that are popular differ by country,
but there seems be a trend of scammers using hooks that are most likely to
yield results in their target locales. For example, high rates of gambling
related scams in Indonesia is correlated with a rapid rise in the illegal
gambling industry in the country, with one PPTAK (Indonesian Financial
Transaction Reports and Analysis Center) spokesperson noting that the turnover
in this industry grew from 57 trillion Rp in 2021 to 81 trillion Rp in 2022
Bhwana2023-rp. Similarly, we note that the high rate of jobs scams in South
Africa is correlated with South Africa having one of the highest unemployment
rates in the world at 32.1%, signifying that scammers are preying on a large
market of job-seekers in the market Desiree_Manamela2024-nz.

####  5.1.3 Scams go severely under-reported, creating ecosystem-wide blind
spots.

Since globally only about half the scams are reported to authorities, the
current national level estimates of scam victimization and prevalence are most
likely highly underestimated (see Figure 5). This trend of under-reporting is
especially pronounced in countries like Indonesia, South Africa and Mexico,
where more than 1 in 5 users reported being unable to report scams to
authorities due to unclear reporting mechanisms. As we discuss further in next
section, generating public awareness of reporting tools and mechanisms
available to the general public could be a productive area of intervention.

The low effective rates of enforcement against scams and low likelihood of
loss recovery may also play a role in discouraging victims from reporting
scams to authorities Dolgin2023-bh. For example, the 2023 State of Scams in
Asia report by Global Anti Scam Alliance, a cross-stakeholder body, found that
there was a positive correlation between the recovery of financial losses and
people’s willingness to report scams to authorities Global-Anti-Scam-
Alliance2023-fb. Another recent report by Third Way, a public policy think
tank, estimated that in the US the enforcement rate for reported incidents to
the Internet Crime Complaint Center (IC3) was 0.3% Eoyang2018-wk. Governments
and central banks must ensure allocation of more resources to law enforcement
agencies, a streamlined reporting mechanism, and ensure that people have
access to proper redressal mechanisms .

###  5.2 Addressing the Scam Problem: Potential Interventions

Scam victimization erodes consumer trust, jeopardizes businesses and harms the
broader economy. A multi- faceted targeted intervention approach is necessary
to develop and assess the effectiveness of various intervention approaches. As
we show in Section 4.3, internet based communication methods play a critical
mediating role in victimization. Prior work in this area has classified scams
along the following broad categories ([DeLiema2023-yo]):

  1. 1.

Opportunity-based scams: typically involves some monetary / social / emotional
reward based hook.

  2. 2.

Threat-based scams: typically involving threats of negative consequences and
blackmail

  3. 3.

Consumer purchase scam: typically happens during online shopping or
marketplace interactions.

  4. 4.

Phishing scams: often involves techniques like impersonation of a
person/entity that victims trust. This is then parlayed to elicit either money
or sensitive information from the victim.

As we show in Section 4.2, the top scam types remain fairly consistent across
surveyed countries; the most popular scam channels were shopping (consumer
purchase scam), job/investment (opportunity-based scam) and impersonation-
based scams (phishing, romance or friends/family impersonation scams). This
bodes well for an approach where we can target and educate users about the
underlying manipulation techniques used in these scams, strengthen the
resilience of such channels themselves, and raise overall awareness. For
countries with outlier patterns, like Indonesia (high victimization from
lottery scams), South Africa (high victimization from job-based scams), and
South Korea (impersonation and romance scams), it would make sense to take a
more white glove approach where Public-Private sector collaboration is called
for to target and curb these trends.

##### Interventions to strengthen user resilience

Recognizing that users are often the weakest link in the scam ecosystem
Baral2019-ab, we must prioritize strategies to enhance their resilience. A
multi- faceted approach, akin to a “swarm the problem” strategy, is necessary
to develop and assess the effectiveness of various intervention approaches.
This area is ripe with opportunities to borrow from intervention tool kits
developed and tested against other forms of online harms like misinformation
Kozyreva2022-to, for example:

  1. 1.

General Awareness Campaigns: Public awareness campaigns help raise awareness
of common scam tactics and warning signs, and can theoretically help the
broader public to recognize and avoid potential threats Miller1983-md. Having
some pre-existing familiarity with scam type has been shown to help
individuals better manage risks, but while general awareness campaigns are one
of the most prominent intervention strategy deployed by stakeholders, its
efficacy in bolstering user resiliency is mixed and unproven Downs2006-wz
Jensen2024-qh. There is some research showing efficacy of such campaigns in
raising awareness about reporting channels, and in that capacity this could be
a tool of interest for stakeholders to potentially alleviate the problem of
under-reporting we noticed across most countries Jeremy-Burke-Francisco-Perez-
Arce-Christine-Kieffer-Robert-Mascio-Gary-Mottola-Olivia-Valdes2021-xv.

  2. 2.

Behavioral Nudges and Credibility Cues: Embedding lightweight behavioral
nudges might be a good low-friction way of guiding users towards safer online
practices Pennycook2021-gm. For instance, research has shown that using
attention prompts to verify if someone is receiving or sending money on a fast
payment app can significantly reduce the risk of falling victim to scams where
scammers rely on the user not paying attention and sending money instead of
receiving it Jha2022-dp. Furthermore, since impersonation based scams are
prevalent across all surveyed countries and users rely on visual trust
indicators to assess credibility online Jakobsson2007-wx, another intervention
worth exploring more broadly here would be providing users with additional
credibility cues (e.g. verification check marks) for verified accounts across
platforms. As we show in Figure 3, since most scams begin from platforms like
social media, email, phone call and text messages, adding positive credibility
cues (like verification check marks) or negative credibility cues (like
potential spam label) could forewarn individuals and help guide them towards
making safer interaction decisions online.

  3. 3.

Inoculation-Styled Interventions: Scammers often rely on a handful of
manipulation techniques to trick their victims into falling for scams. Like
vaccines, these interventions expose users to weaker forms of these techniques
in a controlled environment to help them build immunity or “mental antibodies”
to these real-world threats McGuire1961-yv. Prior research has attempted to
map out user journeys of victims and understand what manipulation techniques
are at play, particularly for romance scams Whitty2013-gb. Building on this,
future research could replicate such mapping to the 3 most popular scam
journeys across countries: consumer purchases online, opportunity-based scams
(job/investment) and impersonation-based scams (phishing, romance and
family/friend impersonation). Users can then be inoculated against these
manipulation techniques and tactics. For example, Robb & Wendel 2023 showed
that there is strong evidence that inoculation techniques increased users’
ability to detect SSN scam emails (government impersonation) as compared to
general tips about scams autociteRobb2023-gz.

  4. 4.

Digital and Financial Literacy-based Interventions: Since scammers prey on
lack of digital & financial literacy among new and habitual internet users,
deploying low-cost comprehensive intervention programs that equip individuals
with the knowledge and skills to navigate the digital landscape safely,
critically evaluate information, and make informed decisions would go a long
way as a preventative measure in this space McGrew2023-uh. Research has shown
that educational interventions can reduce susceptibility to financial fraud
pitches Burke2022-wv. While digital literacy is known to enhance resilience
Graham2017-he, the field lacks interventions comparable to those addressing
misinformation Kozyreva2022-to. Such tools could be powerful interventions
against opportunity-based scams, particularly those that involve investment
schemes preying on individuals with low financial literacy.

###  5.3 Scope for Future Research

##### Deepening research to understand scam victimization and barriers to
reporting

To effectively combat scams, we need a more comprehensive understanding of the
phenomenon. While quantitative data, such as that presented in this study,
helps to gauge the scale of victimization, we also need equally powerful
qualitative research to better understand the issues uncovered in this study,
particularly in less affluent countries. In-depth qualitative research,
particularly in the following areas, could help tailor effective interventions
and policies:

  1. 1.

Understanding Lived Experiences of Scam Victims: Scam victimization leaves
behind profound and lasting scars. Beyond immediate impact like financial
hardships, and emotional repercussions like feelings of shame, anger, and
betrayal, it can lead to long-term instabilities and victims socially
isolating themselves resulting from a loss of trust in those around them
Cross2018-ht, Button2014-bj, Whitty2016-gm. But while qualitative research
related to lived experiences of victims in more affluent countries has been
insightful, experiences of victims in non-affluent countries largely remains a
blind spot. Findings from Section 4.1 in this study underscore an urgent need
for such studies as GNI per-capita and scam victimization rates were strongly
negatively correlated. Understanding victim’s coping strategies, whether
seeking support, changing their behavior, or even withdrawing, is critical for
developing effective support from industry and policymakers.

  2. 2.

Understanding Vulnerability and Resilience: As we discuss in Section 4.2,
while top scam types remain fairly consistent across surveyed countries, scam
exposure and victimization rates vary across countries. This suggests that
there is a complex interplay of factors contributing to vulnerability and
resilience among populations across these countries. Prior research has
identified demographic factors (like income, age, education), psychological
traits and risky online behavior as key contributors to victimization
Vitak2018-kr, Modic2018-ss, Whitty2020-gw, Norris2019-fi, Modic2013-wn. We
find that while demographic factors like income and internet access are linked
to scam victimization, financial losses stemming from victimization is more
common in less affluent countries, even when controlling for internet access.
This suggests that factors such as cultural attitudes toward risk,
financial/digital literacy, and access to information and support could also
play a role in resilience, warranting further research.

  3. 3.

Identifying Barriers to Scam Reporting in countries with high under-reporting:
The under-reporting of scams, particularly in less affluent countries where
individuals lack awareness of or access to reporting mechanisms, hinders
efforts to understand and address the issue, as highlighted by our results
(see Figure 5). Prior research has shown that shame, stigma, and fear of
retaliation or judgment often prevent victims from seeking help or reporting
scams, particularly in romance scams Havers2024-ah. Distrust in authorities
and victim-blaming further deter victims from reporting Button2009-wz.
Complexity of reporting processes can also sometimes be a barrier, especially
considering victims’ post-victimization state of mind Button2013-mz. It is
likely that this problem can be addressed with increased focus on qualitative
research in such areas to understand underlying factors leading to high under-
reporting in these countries.

##  6 Conclusion

Through a nationally representative, multi-country, scams-focused survey, the
present study addresses a substantial gap in our understanding of the global
scams landscape. Our findings substantiate recent theories about the role of
real-time payments in facilitating scams’ proliferation, and highlight the
value of consistent scams reporting across countries to help us understand
regional issues in context. By analyzing data from a diverse range of
economies and cultures, we contribute important insights for researchers,
practitioners, and policymakers in online fraud and scams prevention.

## Authors

Mo Houtti is a Student Researcher on the Trust & Safety Research Team at
Google.

Abhishek Roy is a Staff UX Researcher on the Trust & Safety Research Team at
Google.

Venkata Narsi Reddy Gangula is a Senior UX Researcher on the Trust & Safety
Research Team at Google.

Ashley Marie Walker is a Senior UX Researcher on the Trust & Safety Research
Team at Google.

## Disclosures

This research was funded in it’s entirety by Google. Google is a member of the
Global Anti-Scam Alliance, a coalition body of over 100 organizations,
including governments, law enforcement, consumer protection, financial
authorities and services, social media, internet service providers and
cybersecurity professionals.


# On the Curling Number of Certain Graphs

Johan Kok  Tshwane Metropolitan Police Department
City of Tshwane, Republic of South Africa.
E-mail<EMAIL_ADDRESS>Sudev Naduvath Department of Mathematics
Vidya Academy of Science & Technology
Thalakkottukara, Thrissur, India.
E-mail<EMAIL_ADDRESS>Chithra Sudev Naduvath Mana, Nandikkara
Thrissur, India.
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce the concept of curling subsequence of simple,
finite and connected graphs. A curling subsequence is a maximal subsequence
$C$ of the degree sequence of a simple connected graph $G$ for which the
curling number $cn(C)$ corresponds to the curling number of the degree
sequence per se and hence we call it the curling number of the graph $G$. A
maximal degree subsequence with equal entries is called an identity
subsequence. The number of identity curling subsequences in a simple connected
graph $G$ is denoted $ic(G)$. We show that the curling number conjecture holds
for the degree sequence of a simple connected graph $G$ on $n\geq 1$ vertices.
We also introduce the notion of the compound curling number of a simple
connected graph $G$ and then initiate a study on the curling number of certain
standard graphs like Jaco graphs and set-graphs.

Keywords: Curling number, curling subsequence, identity curling subsequence,
compound curling number, Rasta graph.

AMS Classification Numbers: 05C07, 05C20, 05C38, 05C70, 05C75

##  1 Introduction

For general notation and concepts in graph theory, we refer to [2, 5, 7]
and[11]. All graphs mentioned in this paper are simple, connected finite
graphs unless mentioned otherwise.

The notion of curling number of a sequence of integers is introduced in [3] as
follows.

Let $S=S_{1}S_{2}S_{3}\ldots S_{n}$ be a finite string. Write $S$ in the form
$XYY\ldots Y=XY^{k}$, consisting of a prefix $X$ (which may be empty),
followed by $k$ copies of a non-empty string $Y$. This can be done in several
ways. Pick one with the greatest value of $k$ . Then, this integer $k$ is
called the curling number of $S$ and is denoted by $cn(S)$.

The Curling Number Conjecture (see [4]) states that if one starts with any
finite string, over any alphabet, and repeatedly extends it by appending the
curling number of the current string, then eventually one must reach a $1$.

More studies on curling number of integer sequences have been done in [3] and
and [4]. Motivated from these studies, we extend the concepts of curling
number of sequences to the degree sequences of graphs and hence establish some
interesting results in this area.

In our present discussion, we slightly amend our notation and definition of
the curling number so that we are able to introduce a new notion called
compound curling number of a simple connected graph $G$, as explained below.

######  Definition 1.1.

Given an initial finite non-empty sequence
$S_{0}=(a_{1},a_{2},a_{3},\ldots,a_{n})$, $a_{i}\in\mathbb{N}_{0}$, let
$\epsilon$ be the empty subsequence. If we partition the sequence $S_{0}$ into
two subsequences, say $X,Y$, we can write $S_{0}$ as the string $S_{0}=X\circ
Y$. Hence, we can write sequence $S_{0}$ as a string of subsequences
$S_{0}=\epsilon\circ X_{1}^{k^{*}_{1}}$ or $S=X_{1}^{k_{1}}\circ
X_{2}^{k_{2}}\circ\ldots\circ X_{l}^{k_{l}}$, where
$X_{i}^{k_{i}}\neq\epsilon$, $1\leq i\leq l,~{}l\geq 2$.

Henceforth, the initial sequence $S_{0}$ will be considered to be finite and
non-empty unless mentioned otherwise.

######  Definition 1.2.

If after re-arrangement of the entries of a general sequence $S$ to obtain a
specific initial sequence $S_{0}$, we can write $S_{0}=\epsilon\circ
X_{1}^{k^{*}_{1}}$ or $S=X_{1}^{k_{1}}\circ X_{2}^{k_{2}}\circ\ldots\circ
X_{l}^{k_{l}}$, where $X_{i}^{k_{i}}\neq\epsilon$, $1\leq i\leq l,l\geq 2$ and
$k_{1}\leq k_{2}\leq k_{3}\leq\ldots\leq k_{l}$ and $k_{l}$ a maximum over all
possible subsequences, we define the curling number of
$cn(S_{0})=\max\\{k^{*}_{1},k_{l}\\}$.

The term $X^{k_{i}}_{i}$, explained in Definition 1.2, is called a string
factor of the initial sequence $S_{0}$.

Note that if $S_{0}$ has $n$ entries and is written $S_{0}=\epsilon\circ
X^{k^{*}_{1}}_{1}$ then $k^{*}_{1}\in\\{1,n\\}$. Also note that for the
purposes of determining the curling number of an initial sequence $S_{0}$ we
only have to write either $S_{0}=\epsilon\circ X^{k^{*}_{1}}_{1}$ or
$S_{0}=X^{k_{1}=1}_{1}\circ X^{k_{2}}_{2}$ for $\max k_{2}$ over all possible
subsequences. It follows easily that $cn(S_{0})=cn(X^{k_{2}}_{2})$ for $\max
k_{2}$ over all possible subsequences.

##  2 Curling Sub-Sequences of Graphs

Let us first introduce the notion of a curling subsequence of the degree
sequence of a simple connected graph as follows.

######  Definition 2.1.

A curling subsequence of a simple connected graph $G$ is defined to be a
maximal subsequence $C$ of the well-arranged degree sequence of $G$ such that
$cn(C)=\max\\{cn(S_{0})\\}$ for all possible initial subsequences $S_{0}$.

######  Definition 2.2.

The curling number of a graph $G$ is the curling number of a curling
subsequence $C$ of $G$. That is, $cn(G)=cn(C)$, where $C$ is a curling
subsequence of $G$.

Note that a graph $G$ can have a number of curling subsequences. Hence, we
have the following notions.

######  Definition 2.3.

The number of curling subsequences in a given graph $G$ is called the curling
index of $G$ and is denoted by $\vartheta(G)$.

######  Definition 2.4.

A maximal degree subsequence with equal entries is called an identity
subsequence. An identity subsequence can be a curling subsequence, the number
of identity curling subsequences found in a simple connected graph $G$ is
denoted $ic(G)$.

An illustration to the above notions is given below.

######  Illustration 2.5.

Consider the graph $G$ with degree sequence $(3,5,3,3,5,5,6)$. After re-
arrangement of the entries, we can consider the four possible initial degree
sequences $S_{0}=(6,3,3,3,5,5,5)$ or $S_{0}=(6,5,5,5,3,3,3)$ or
$S_{0}=(6,3,5,3,5,3,5)$ or $S_{0}=(6,5,3,5,3,5,3)$. We can identify the two
identity curling subsequences $(3,3,3)$ and $(5,5,5)$ as well as the two
additional curling subsequences $(3,5,3,5,3,5)$ and $(5,3,5,3,5,3)$. Since
$(3,3,3)=\epsilon(3)^{3}$, we have $cn((3,3,3))=3$. Similarly,
$(5,5,5)=\epsilon(5)^{3},cn((5,5,5))=3$,
$(3,5,3,5,3,5)=\epsilon(3,5)^{3},cn((3,5,3,5,3,5))=3$ and
$(5,3,5,3,5,3)=\epsilon(5,3)^{3},cn((5,3,5,3,5,3))=3$. Therefore $cn(G)=3$.
Note that $(6)$ is an identity subsequence as well.

Invoking the above definitions, we establish the following theorem on the
number of curling subsequences of a given graph.

######  Theorem 2.6.

The number of curling subsequences of a simple connected graph $G$ is given by

$\vartheta(G)=\begin{cases}1&\text{if}~{}ic(G)=1\\\
ic(G)+ic(G)!&\text{otherwise}\end{cases}$

###### Proof.

If $ic(G)=1$, then, clearly by the definition of curling subsequence, $G$ can
have exactly one curling subsequence. That is, $\vartheta(G)=1$ if $ic(G)=1$

Then, for a positive integer $l\geq 2$, assume that the graph $G$ has $l$
identity curling subsequences, say
$S_{1}=(\alpha_{1},\alpha_{1},\alpha_{1},\ldots,\alpha_{1})$,
$S_{2}=(\alpha_{2},\alpha_{2},\alpha_{2},\ldots,\alpha_{2},)$,
$S_{3}=(\alpha_{3},\alpha_{3},\alpha_{3},\ldots,\alpha_{3})$ and proceeding
like this up to $S_{l}=(\alpha_{l},\alpha_{l},\alpha_{l},\ldots,\alpha_{l})$,
each of these subsequences containing $k\geq 2$ terms.

Now, we can construct exactly $ic(G)\,!=l\,!$ sequences of the form
$(\alpha_{i},\alpha_{j},\alpha_{m},\ldots,\\\ \alpha_{s})$ containing $l$
entries each of which are distinct. Hence, the degree sequence can be written
as $X^{k_{1}=1}_{1}\circ X^{ic(G)}_{2}$, with
$X^{ic(G)}_{2}\in\\{\alpha_{i},\alpha_{j},\alpha_{m},\ldots,\alpha_{s}\\}$,
where each of these $l$ entries are distinct and $1\leq i,j,m,\ldots,s\leq l$.
Therefore, $\vartheta(G)=ic(G)+ic(G)!$ ways. This completes the proof. ∎

The validity of curling number conjecture for a simple, connected graph under
consideration is one of the most interesting problem in this context. The
following theorem establishes the existence of curling number conjecture for a
given graph.

######  Theorem 2.7.

For the degree sequence of a non-trivial, connected graph $G$ on $n$ vertices,
the curling number conjecture holds.

###### Proof.

Consider a graph $G$ on $n$ vertices. If $n=1$, then $G=K_{1}$ and hence
$S_{0}=(0)$, $S_{1}=(0,1)$ so the conjecture holds trivially. Now, let $n\geq
2$. If the degree sequence of $G$ can only be written as $S_{0}=\epsilon\circ
X^{k^{*}_{1}}_{1}$ then $k^{*}_{1}=$ either $1$ or $n$.

Case 1: If $k^{*}_{1}=1$ and the last entry of the degree sequence is 1, then
$S_{1}=S_{0}\circ(1)$, $S_{2}=S_{0}\circ(1,2)$, $S_{3}=S_{0}\circ(1,2,1)$. If
the last entry is larger than 1, then $S_{1}=S_{0}\circ(1)$.

Case 2: If $k^{*}_{1}=n$ the $G$ is a regular graph. So $S_{0}=\epsilon\circ
X^{n}_{2}$. So $S_{1}=\epsilon\circ X^{n}_{2}\circ(n)$, $S_{2}=\epsilon\circ
X^{n}_{2}\circ(n1)$.

Case 3: If the degree sequence can be written as $X^{k_{1}=1}_{1}\circ
X^{k_{2}}_{2}$ for $\max k_{2}$ and $X^{k_{2}}_{2}$ is an identity curling
subsequence, over all possible subsequences, we have that if the entry values
are $\alpha$ and the number of entries is $\alpha$, then
$S_{1}=X^{k_{1}=1}_{1}\circ X^{\alpha}_{2}\circ(\alpha)$,
$S_{2}=X^{k_{1}=1}_{1}\circ X^{\alpha}_{2}\circ(\alpha,\alpha+1)$,
$S_{3}=X^{k_{1}=1}_{1}\circ X^{\alpha}_{2}\circ(\alpha,\alpha+1,1)$. For all
other cases we have that $S_{0}=X^{k_{1}=1}_{1}\circ
X^{k_{2}}_{2}=X^{k_{1}=1}_{1}\circ X^{\beta}_{2},\beta\neq\alpha$. So
$S_{1}=X^{k_{1}=1}_{1}\circ X^{\beta}_{2}\circ(\beta)$,
$S_{2}=X^{k_{1}=1}_{1}\circ X^{\beta}_{2}\circ(\beta,1)$.

Hence, the conjecture holds in all the above three cases. This completes the
proof. ∎

We note that any degree sequence can be re-arranged to be written as a string
of identity subsequences. In view of this fact, we propose the following
definition.

######  Definition 2.8.

If the degree sequences of graphs $G_{1}$ and $G_{2}$ are written as strings
of identity subsequences and the entries in two identity subsequences
$X_{i},X_{j}$ ($X_{i}$ in $G_{1}$ with $s$ entries and $X_{j}$ in $G_{2}$ with
$t$ entries) are equal they are called similar identity subsequences and the
merger of these identity subsequences is the identity subsequence with $s+t$
entries which is denoted, $X_{i}\uplus X_{j}$.

The merger of similar identity subsequences plays an important role when
considering the union of a number of graphs. Hence, if
$G=\bigcup\limits^{m}_{i=1}G_{i}$ and each $G_{i}$ has a similar identity
subsequence say $X_{i}=(\alpha,\alpha,\alpha,\ldots,\alpha)$ containing
$k_{i}$ entries each, then $X_{1}\uplus X_{2}\uplus X_{3}\uplus\ldots\uplus
X_{m}=(\alpha)^{k}$, where $k=\sum\limits^{m}_{i=1}k_{i}$.

If a particular degree sequence of $G_{j}$ does not contain the identity
subsequence $X_{r}$ then it is represented by the empty identity subsequence
$\epsilon$ and $k_{r}=0$.

Invoking the above concepts on similar identity subsequences, we establish the
following theorem on the curling number of a graph which is the union of
finite number of finite graphs.

######  Theorem 2.9.

If a graph $G$ is the union of $m$ simple connected graphs $G_{i};~{}1\leq
i\leq m$ and the respective degree sequences are re-arranged as strings of
identity subsequences, then

$cn(G)=\begin{cases}\max\\{cn(G_{i})\\},&\text{if}~{}X_{i},X_{j}\text{are not
pairwise similar},\\\ \max\sum\limits^{m}_{i=1}k_{i},&\text{for all merger of
similar identity subsequences.}\end{cases}$

###### Proof.

  1. (i)

If $X_{i},X_{j}$ are not pairwise similar for any values of $i$ and $j$, then
any identity subsequence can be written as $(\alpha)^{k_{i}\geq 1}$. Hence, a
$\max k_{j}$ exists for some $j$ thus,
$cn(G)=cn(\bigcup\limits^{m}_{i=1}G_{i})=\max\\{cn(G_{i}):1\leq i\leq m\\}$.

  2. (ii)

After all mergers of similar identity subsequences, re-arrange the degree
sequence of $G$ in such a way that it can be written as
$(\alpha_{1})^{k_{1}}\circ(\alpha_{2})^{k_{2}}\circ)\alpha_{3})^{k_{3}}\circ\ldots\circ(\alpha_{l})^{l=\max\sum\limits^{m}_{i=1}k_{i}}$.
From the Definition 1.2, it follows that,
$cn(G)=cn(\bigcup\limits^{m}_{i=1}G_{i})=\max\sum\limits^{m}_{i=1}k_{i}$.

∎

###  2.1 Compound Curling Number of a Graph

As we have stated earlier, any degree sequence of a graph $G$ can be written
as a string of identity curling subsequences. In view of this fact, we
introduce the concept of the compound curling number of a graph $G$ as
follows.

######  Definition 2.10.

Let the degree sequence of the graph $G$ be written as a string of identity
curling subsequences say, $X^{k_{1}}_{1}\circ X^{k_{2}}_{2}\circ
X^{k_{3}}_{3}\ldots\circ X^{k_{l}}_{l}$. The compound curling number of $G$,
denoted $cn^{c}(G)$ is defined to be, $cn^{n}(G)=\prod\limits^{l}_{i=1}k_{i}$.

Next, let us illustrate the curling number and compound curling number of
certain fundamental standard graphs.

  1. 1.

The degree sequence of $K_{n}$ is $(n-1,n-1,n-1,\ldots,n-1)$, consisting of
$n$ terms. Hence, $S_{0}=\epsilon\circ(n-1)^{n}$. Therefore, $cn(K_{n})=n$ and
$cn^{c}(K_{n})=n$.

  2. 2.

The degree sequence of $K_{m,n}$ is
$(\underbrace{m,m,\ldots,m,}_{n-terms}\underbrace{n,n,\ldots,n}_{m-terms})$.
Therefore, $S_{0}=(m)^{n}\circ(n)^{m}$. Hence, $cn(K_{m,n})=\max\\{m,n\\}$.
Similarly, for $K_{n,n}$, we have $S_{0}=\epsilon\circ(n)^{2n}$ and hence
$cn(K_{n,n})=2n$. Moreover, the compound curling number of $K_{m,n}$ is $mn$
and that of $K_{n,n}$ is $2n$ itself.

  3. 3.

For a path $P_{n}$ on $n$ vertices, we have $S_{0}=(1)^{2}\circ(2)^{n-2}$. If
$n=2$, then $S_{0}=\epsilon\circ(1)^{2}$ and if $n=3$, then
$S_{0}=(1)^{2}\circ(2)^{1}$. Hence, $cn(P_{n})=2$ for $n=2,3$ and
$cn(P_{n})=n-2$ if $n\geq 4$. Also, the compound curling number of $P_{n}$ is
given by $cn^{c}(P_{n})=2$ for $n=2,3$ and $cn^{c}(P_{n})=2(n-2)$.

  4. 4.

For a cycle $C_{n}$, we have $S_{0}=\epsilon\circ(2)^{n}$ and hence
$cn(C_{n})=n=cn^{c}(C_{n})$.

  5. 5.

A wheel graph is defined as $W_{n}=C_{n-1}+K_{1}$. The degree sequence of
$W_{n}$ is $S_{0}=(n-1)^{1}\circ(3)^{n-1}$. Hence, the curling number and the
compound curling number of the wheel graph $W_{n}$ are $n-1$.

  6. 6.

A ladder graph $L_{n}$ is the graph defined by $L_{n}=P_{n}\times P_{2}$,
where $n\geq 2$. If $n=2$, then $L_{2}=C_{4}$, and this case we have already
discussed. Therefore, for $n>2$, in $L_{n}$, there are $2n$ vertices of which
$4$ vertices have degree $2$ and all other vertices have degree $3$. That is,
the degree sequence of $L_{n}$ is $(2)^{4}\circ(3)^{2n-4}$. Therefore,
$cn(L_{n})=2(n-2)$ and $cn^{c}(L_{n})=8(n-2)$.

  7. 7.

An $m$-regular graph of girth $n$ with minimum possible vertices is called an
$(m,n)$-cage. $(2,n)$-cage is an $n$-cycle and hence its curling number is
$n$, $(m,3)$-cage is a complete graph $K_{m+1}$ and hence its curling number
is $m+1$ and the $(m,4)$-cage is the complete bipartite graph $K_{m,m}$ and
hence its curling number is $2m$. For a general (m,n)-cage $G$, the number of
vertices required is denoted by $f(m,n)$ and is determined as
$f(m,n)=\begin{cases}\frac{m(m-1)^{r}-2}{m-2},&\text{if}~{}r=\frac{n-1}{2},\\\
\frac{2(m-1)^{r}-2}{m-2},&\text{if}~{}r=\frac{n}{2}.\end{cases}$ Therefore,
the curling number of an $(m,n)$-cage is $f(m,n)$. It can also be noted that
the compound curling number of a cage graph is equal to its curling number.

Invoking the above definitions, we arrive at the following results.

######  Proposition 2.11.

The compound curling number of any regular graph $G$ is equal to its curling
number.

###### Proof.

Let $G$ be an $r$-regular graph on $n$ vertices. Then, the proof is obvious
from the fact that the degree sequence of $G$ is $\epsilon\circ(r)^{n}$. ∎

######  Proposition 2.12.

If the degree sequence of two graphs $G$ and $H$ can be written as a string of
identity curling subsequences, $X^{k_{1}}_{1}\circ X^{k_{2}}_{2}\circ
X^{k_{3}}_{3}\circ\ldots\circ X^{k_{t}}_{t}$, and $Y^{l_{1}}_{1}\circ
Y^{l_{2}}_{2}\circ Y^{l_{3}}_{3}\circ\ldots\circ X^{k_{i}}_{i}\circ\ldots\circ
Y^{l_{s}}_{s}$, respectively, then the compounding curling number of the union
of $G$ and $H$ is given by $cn^{c}(G\cup H)=cn^{c}(G)\prod\limits_{1\leq j\leq
s,~{}j\neq i}Y^{l_{j}}_{j}+cn^{c}(H)\prod\limits_{1\leq j\leq t,~{}j\neq
i}X^{k_{j}}_{j}$.

###### Proof.

The proof of this proposition is very clear and straight forward. ∎

Let $n=\sum\limits_{i=n}^{l}t_{i}$, where each $t_{i}\in\mathbb{N}$ such that
$t_{i}>t_{i+1},~{}\forall~{}i=1,2,\ldots,l$ and $t_{l}>1$. The general
$l$-term summand set is the set $\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$.

By a minimal graph with respect to a given property, we mean a graph with
minimum order and size satisfying that property. Next, using this terminology,
let us introduce the terminology of some connected graphical embodiment for
general $l$-term summand sets.

We now describe the construction of the minimal connected graphical embodiment
called the Rasta graph, denoted by $G^{(l)}$, for a general $l$-term summand
set which satisfies the condition $cn^{c}(G^{(l)})=\prod\limits_{1\leq i\leq
l,~{}i\neq 3}t_{i}$, and $cn^{c}(G^{(3)})=t_{2}(t_{1}+t_{3})$ as follows.

### Constructing a Rasta graph

Consider a $t$-term summand set $\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ with
$t_{1}>t_{2}>t_{3}>\ldots>t_{l}>1$. Then, a Rasta graph can be constructed in
the following steps.

  1. S-1:

Consider $t_{1}$ vertices in the left column (first column) and $t_{2}$
vertices in the right column (second column) and construct $K_{t_{1},t_{2}}$.

  2. S-2:

Add the third column of $t_{3}$ vertices and add the edges of
$K_{t_{2},t_{3}}$

  3. S-3:

Repeat Step 2 iteratively up to $t_{l}$

  4. S-4:

Exit.

We note that a Rasta graph is the underlying graph of a directed graph
mentioned in the following definition.

######  Definition 2.13.

For a $l$-term summand set $\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ with
$t_{1}>t_{2}>t_{3}>\ldots>t_{l}>1$ define the directed graph $G^{(l)}$ with
vertices $V(G^{(l)})=\\{v_{i,j}:1\leq j\leq t_{i},1\leq i\leq l\\}$ and the
arcs, $A(G^{(l)})=\\{(v_{i,j},v_{(i+1),m}):1\leq i\leq(l-1),1\leq j\leq t_{i}$
and $1\leq m\leq t_{(i+1)}\\}$.

The following lemma establishes a relation between the compound curling
numbers of a complete bipartite graph and a connected graphical embodiment of
a $2$-term summand set.

######  Lemma 2.14.

For a $2$-term summand set $\\{a,b\\}$, where $a>b$, compound curling number
of the minimal connected graphical embodiment $G^{(2)}$ is given by
$cn^{c}(G^{(2)})$ is equal to that of a complete bipartite graph $K_{a,b}$.

###### Proof.

Clearly $G^{(2)}$ must have $a+b$ vertices. Connecting each vertex $v_{1,j}$,
where $1\leq j\leq a$ to each vertex $v_{1,m}$ $1\leq m\leq b$ ensures the
minimal connected graphical embodiment $G^{(2)}$ such that
$d_{G}^{(2)}(v_{1,j})=b$, $1\leq j\leq a$. Hence, a factor of the degree
sequence string is given by $(b)^{a}$. It follows that this minimal graphical
embodiment resulted in the graph $G^{(2)}=K_{a,b}$. Hence, the degree sequence
can be written as the string $(b)^{a}\circ(a)^{b}$. Therefore,
$cn^{c}(G^{(2)})=ab=cn^{c}(K_{a,b})$. ∎

The compound curling number of the Rasta graph corresponding to an $l$-term
summand set is determined in the following theorem.

######  Theorem 2.15.

For $n\in\mathbb{N}$ and any $l$-term summand set of $n$ say
$\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ with
$t_{1}>t_{2}>t_{3}>\ldots>t_{l}>1$, the corresponding Rasta graph is a minimal
connected graphical embodiment $G^{(l)}$ such that
$cn^{c}(G^{(l)})=\prod\limits_{i=1,\,i\neq 3}^{l}t_{i}$ and
$cn^{c}(G^{(3)})=t_{2}(t_{1}+t_{3})$.

###### Proof.

For a $2$-term summand set $\\{t_{1},t_{2}\\}$, where $t_{1}>t_{2}$, the Rasta
graph is $K_{t_{1},t_{2}}$. Clearly, the number of vertices and edges are
minimal to provide the degree sequence
$(\underbrace{t_{2},t_{2},\ldots,t_{2}}_{t_{1}-entries},\underbrace{t_{1},t_{1},t_{1},\ldots,t_{1}}_{t_{2}-entries})$
(See Lemma 2.14). Hence, the degree sequence can be written as the string
$(t_{2})^{t_{1}}\circ(t_{1})^{t_{2}}$ so
$cn^{c}(K_{t_{1},t_{2}})=\prod\limits_{i=1}^{2}t_{i}$.

For a $3$-element summand set $\\{t_{1},t_{2},t_{3}\\}$, the corresponding
Rasta graph has degree sequence can be written as the string of identity
subsequences
$S_{0}=(t_{2})^{t_{1}}\circ(t_{1}+t_{3})^{t_{2}}\circ(t_{2})^{t_{3}}=(t_{1}+t_{3})^{t_{2}}\circ(t_{2})^{(t_{1}+t_{3})}$.
Therefore, $cn^{c}(G^{(3)})=t_{2}(t_{1}+t_{3})$.

Finally, consider an general $l$-term summand set
$\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ of an integer $n$, where ${l\neq 3}$
such that $t_{1}>t_{2}>t_{3}>\ldots>t_{l}>1$ and $l\geq 2$. Clearly, the Rasta
graph corresponding to this $l$-term summand set is a minimal graph whose
degree sequence is
$(t_{2})^{t_{1}}\circ(t_{1}+t_{3})^{t_{2}}\circ(t_{2}+t_{4})^{t_{3}}\circ\ldots,\circ(t_{l-2}+t_{l})^{t_{(l-1)}}\circ(t_{l-1})^{t_{l}}$.
Also, we have $t_{1}>t_{2}>t_{3}>\ldots>t_{l}>1$. Therefore, the result
$cn^{c}(G^{(l)})=\prod\limits_{i=1,\,i\neq 3}^{l}t_{i}$, follows. ∎

Let us now introduce the notion of the maximal $l$-term summand set in respect
of $n\in\mathbb{N}$ as follows.

######  Definition 2.16.

Let $n\geq$ be a positive integer such that
$n=\sum\limits_{i=1}^{l}t_{i},~{}\forall\,t_{i}\in\mathbb{N}$. The maximal
$l$-term summand set of $n\geq 3$ is the $l$-term summand set
$\\{t_{1},t_{2},t_{3},\ldots t_{l}\\}$ such that $t_{i+1}>t_{i}$ for
$i=1,2,3,\ldots,l$ for which the product $\prod\limits_{i=1}^{l}t_{i}$ is
maximum.

A maximal $2$-term summand set for a positive integer $n$ is determined in the
following lemma.

######  Lemma 2.17.

Let $n$ be a positive integer greater than or equal to $3$. Then, the maximal
$2$-term summand set for $n$ is
$\\{\lfloor\frac{n}{2}\rfloor+1,\lfloor\frac{n}{2}\rfloor\\}$.

###### Proof.

By induction, it can easily be followed that,
$\lfloor\frac{n}{2}\rfloor(\lfloor\frac{n}{2}\rfloor+1)>(\lfloor\frac{n}{2}\rfloor-i)((\lfloor\frac{n}{2}\rfloor+1)+i)=\lfloor\frac{n}{2}\rfloor(\lfloor\frac{n}{2}\rfloor+1)-i-i^{2}$
and this completes the proof. ∎

We observe that, for a sufficiently large $n$, a maximal $3$-term summand set
$\\{t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3}\\}$, such that
$t^{\prime}_{1}>t^{\prime}_{2}>t^{\prime}_{3}$, can be obtained from a maximal
$2$-term summand set by applying Lemma 2.17 to $t_{1}$. If $n$ is sufficiently
large, the maximal $4$-term summand set can be obtained by applying Lemma 2.17
to $t^{\prime}_{i}=\max\\{t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3}\\}$ for
which a maximal $4$-term summand set is defined and the process can be
repeated iteratively.

In view of the above concept, we have the following result.

######  Corollary 2.18.

There exists a maximum value $l^{*}\geq 4$ for which we find a defined
$l$-term summand set with respect to the given positive integer $n\geq 3$, has
corresponding Rasta graph $G^{(l^{*})}$ with $cn^{c}(G^{(l^{*})})$ is maximum
over all defined $l$-term summand sets.

###### Proof.

Consider any $l$-term summand set with respect to a positive integer $n\geq
3$, say $\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$. Apply Lemma 2.17 to
$t^{\prime}_{j}=\max\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ for which a maximal
($l+1$)-term summand set is defined. Now, we have
$cn^{c}(G^{(l)})=\prod\limits_{i=1,\,i\neq
3}^{l}t_{i}=(\prod\limits_{i=1,\,i\neq
3,j}^{l}t_{i})t_{j}=(\prod\limits_{i=1,\,i\neq 3,j}^{l}t_{i})(a+b)$, where
$\\{a,b\\}$ is the maximal $2$-term summand set of $t_{j}$. Clearly,
$(\prod\limits_{i=1,\,i\neq 3,j}^{l}t_{i})(a+b)<(\prod\limits_{i=1,\,i\neq
3,j}^{l}t_{i})(ab)$. If a ($l$ \+ 1)-term summand set is not defined, then the
set $\\{t_{1},t_{2},t_{3},\ldots,t_{l}\\}$ gives the required result. Repeat
same procedure until the final $l^{*}$-term summand set is found through
exhaustion. ∎

An illustration to maximal summand sets for a given positive integer $n$ and
the compound curling number of the corresponding Rasta graphs is given below.

######  Illustration 2.19.

For $n=30$, the maximal $2$-term summand set is $\\{16,14\\}$ and the maximal
$3$-term summand set is $\\{14,9,7\\}$, the maximal $4$-term summand set is
$\\{9,8,7,6\\}$, the maximal $5$-term summand set is $\\{8,7,6,5,4\\}$ and the
maximal $6$-term summand set is $\\{8,7,6,4,3,2\\}$. No other maximal $l$-term
summand sets are defined for $n=30$. The degree sequence of corresponding
Rasta graph $G^{(6)}$ can be written as the string
$S_{0}=(7)^{8}\circ(14)^{7}\circ(11)^{6}\circ(9)^{4}\circ(6)^{3}\circ(3)^{2}$.
Hence, we have $cn^{c}(G^{(6)})=8064$, $cn^{c}(G^{(5)})=6720$,
$cn^{c}(G^{(4)}=3024$, $cn^{c}(G^{(3)})=189$ and $cn^{n}(G^{(2)})=224$.

######  Corollary 2.20.

The compound curling number of the Rasta graph $G^{(3)}$ corresponding to a
$3$-term summand set is the minimum among the compound curling numbers of the
Rasta graphs $G^{(l)}$ corresponding to all $l$-term summand sets of $n$.

###### Proof.

We only have to show that $cn^{n}(G^{(2)})>cn^{n}(G^{(3)})$. For a maximal
$2$-term summand set $\\{a,b\\}$ we have $a-b\leq 2$. Hence if
$\\{a_{1},a_{2}\\}$ is the maximal $2$-term summand set of $a$, the maximal
$3$-term summand set is given by $\\{b,a_{1},a_{2}\\}$. We have that
$cn^{c}(G^{(2)})=ab=(a_{1}+a_{2})b=a_{1}b+a_{2}b$. We also have that
$cn^{c}(G^{(3)})=a_{1}(b+a_{2})=a_{1}b+a_{1}a_{2}<a_{1}b+a_{2}b$ because
$b>a_{1}$. ∎

In general, there exists no specific relation between the compound curling
number of a graph and its spanning subgraphs. We discuss a relation between
the compound curling numbers of a regular graph and its spanning subgraph in
the following result.

######  Proposition 2.21.

For a $k$-regular graph G on $n\geq 4$ vertices we have $cn^{c}(G-uv)\geq
cn^{c}(G)$.

###### Proof.

For a $k$-regular graph $G$ on $n\geq 4$ vertices, $cn^{c}(G)=n$. Consider any
$G-uv$ and it follows that the corresponding degree sequence can be written as
the string $(k)^{n-2}\circ(k-1)^{2}$. Hence, $cn^{c}(G-uv)=2(n-2)=n+(n-4)\geq
n$. Hence, $cn^{c}(G-uv)\geq cn^{c}(G)$. ∎

Although Proposition 2.21 is splendidly simple it lays the foundation for a
more profound result. We use the default convention that $(x)^{0}=1$ hence,
$(\prod\limits_{i=1}^{n}i)(x)^{0}=\prod\limits_{i=1}^{n}i$.

######  Theorem 2.22.

Consider a graph G with degree string
$(d_{1})^{t_{1}}\circ(d_{2})^{t_{2}}\circ(d_{3})^{t_{3}}\circ\ldots\circ(d_{l})^{t_{l}}$,
$d_{1}>d_{2}>d_{3}>\ldots>d_{l}$. Assume that there exists at least one
identity subsequence $(d_{j})^{t_{j}};~{}t_{j}\geq 2$ with at least two
vertices $u,v$ with corresponding degree $d_{j}$, adjacent. Then, we have

$\begin{cases}cn^{c}(G-uv)\geq cn^{c}(G),&\text{if}~{}t^{*}_{j}\geq
t_{j+1},\\\ cn^{c}(G-uv)<cn^{c}(G),&\text{if}~{}t^{*}_{j}<t_{j+1}.\end{cases}$

with $t_{j}=t^{*}_{j}+2$.

###### Proof.

If the string factor $(d_{j+1})^{t_{j+1}}$ does not exists, add the default
factor $(d_{j+1})^{0}$. Now
$cn^{c}(G)=\prod\limits_{i=1}^{l}t_{i}=(\prod\limits_{i=1,\,i\neq
j,j+1}^{l}t_{i})t_{j}t_{j+1}$ or $(\prod\limits_{i=1,\,i\neq
j,j+1}^{l}t_{i})t_{j}(d_{j+1})^{0}$. The compound curling number of the graph
$G-uv$ is either $(\prod\limits_{i=1}^{l}t_{i})t^{*}_{j}(t_{j+1}+2)$ or
$(\prod\limits_{i=1}^{l}t_{i})t^{*}_{j}+2$, where $i\neq j,j+1$. Hence,
$cn^{c}(G-uv)\geq cn^{c}(G)$ if
$t^{*}_{j}(t_{j+1}+2)\geq(t^{*}_{j}+2)t_{j+1}$, thus if $t^{*}_{j}\geq
t_{j+1}$.

We can prove the converse part also using similar arguments. ∎

##  3 Curling Number of Jaco Graphs

The notion of Jaco graphs was introduced in [8]. In this paper, we use Jaco
graphs of order $1$ which is defined in [8] as follows.

######  Definition 3.1.

A Jaco graph of order $1$, denoted by $J_{\infty}(1)$, is defined as a
directed graph with the vertex set
$V=V(J_{\infty}(1))=\\{v_{i}:i\in\mathbb{N}\\}$, and the arc set
$A(J_{\infty}(1))\subseteq\\{(v_{i},v_{j}):i,j\in\mathbb{N},i<j\\}$ and
$(v_{i},v_{j})\in A(J_{\infty}(1))$ if and only if $2i-d^{-}(v_{i})\geq j$.

The Jaco graph $J_{\infty}(1)$ has the following four fundamental properties.

  1. (i)

$V(J_{\infty}(1))=\\{v_{i}:i\in\mathbb{N}\\}$,

  2. (ii)

if $v_{j}$ is the head of an arc then the tail is always a vertex $v_{i},i<j$,

  3. (iii)

if $v_{k}$, for smallest $k\in\mathbb{N}$ is a tail vertex then all vertices
$v_{l},k<l<j$ are tails of arcs to $v_{j}$,

  4. (iv)

the degree of vertex $k$ is $d(v_{k})=k$.

The family of finite Jaco graphs are those limited to $n\in\mathbb{N}$
vertices by lobbing off all vertices (and arcs) $v_{t},t>n$ and denoted
$J_{n}(1)$. Hence, trivially we have $d(v_{i})\leq i$ for $i\in\mathbb{N}$.

When defined directed graphs like Jaco graphs are considered the study will
apply to the underlying graph. For $J_{n}(1)$ we denote the underlying graph
$J^{*}_{n}(1)$. When the context is clear a Jaco graph could mean the directed
graph, $J_{n}(1)$ or the underlying graph, $J^{*}_{n}(1)$. We now provide the
degree sequences of Jaco graphs, $n\leq 26$ in the defined order of vertex
labelling (indexing) together with the values
$ic(J^{*}_{n}(1)),\vartheta(J^{*}_{n}(1)),cn(J^{*}_{n}(1))$. See the Fisher
algorithm in [8] for the degree sequences.

######  Illustration 3.2.

The degree sequence, number of identity curling subsequences, curling index
and curling number and of first few finite Jaco graphs $J^{*}_{n}(1)$ are
provided in the following table.

$n$ | degree sequence | $ic$ | $\vartheta$ | $cn$
---|---|---|---|---
1 | (0) | 1 | 1 | 1
2 | (1,1) | 1 | 1 | 2
3 | (1,2,1) | 1 | 1 | 2
4 | (1,2,2,1) | 2 | 4 | 2
5 | (1,2,3,2,2) | 1 | 1 | 3
6 | (1,2,3,3,3,2) | 1 | 1 | 3
7 | (1,2,3,4,4,3,3) | 1 | 1 | 3
8 | (1,2,3,4,5,4,4,3) | 1 | 1 | 3
9 | (1,2,3,4,5,5,5,4,3) | 1 | 1 | 3
10 | (1,2,3,4,5,6,6,5,4,4) | 1 | 1 | 3
11 | (1,2,3,4,5,6,7,6,5,5,4) | 1 | 1 | 3
12 | (1,2,3,4,5,6,7,7,6,6,5,4) | 1 | 1 | 3
13 | (1,2,3,4,5,6,7,8,7,7,6,5,5) | 2 | 4 | 3
14 | (1,2,3,4,5,6,7,8,8,8,7,6,6,5) | 2 | 4 | 3
15 | (1,2,3,4,5,6,7,8,9,9,8,7,7,6,6) | 2 | 4 | 3
16 | (1,2,3,4,5,6,7,8,9,10,9,8,8,7,7,6) | 2 | 4 | 3
17 | (1,2,3,4,5,6,7,8,9,10,10,9,9,8,8,7,6) | 2 | 4 | 3
18 | (1,2,3,4,5,6,7,8,9,10,11,10,10,9,9,8,7,7) | 3 | 9 | 3
19 | (1,2,3,4,5,6,7,8,9,10,11,11,11,10,10,9,8,8,7) | 3 | 9 | 3
20 | (1,2,3,4,5,6,7,8,9,10,11,12,12,11,11,10,9,9,8,8) | 3 | 9 | 3
21 | (1,2,3,4,5,6,7,8,9,10,11,12,13,12,12,11,10,10,9,9,8) | 3 | 9 | 3
22 | (1,2,3,4,5,6,7,8,9,10,11,12,13,13,13,12,11,11,10,10,9,8) | 3 | 9 | 3
23 | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,14,13,12,12,11,11,10,9,9) | 3 | 9 | 3
24 | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,14,13,13,12,12,11,10,10) | 3 | 9 | 3
25 | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,15,14,14,13,13,12,11,11,10,9) | 3 | 9 | 3

In the above table $ic,~{}\vartheta$ and $cn$ stands for $ic(J^{*}_{n}(1))$,
$\vartheta(J^{*}_{n}(1))$ and $cn(J^{*}_{n}(1))$ respectively.

From Illustration 3.2, we see that any degree sequence of $J^{*}_{n}(1),n\geq
5$ can be re-arrange to have one or more 3-entry identity curling
subsequence(s) at the tail. For example, the degree sequence of the Jaco graph
$J^{*}_{19}(1)$ can be re-arranged as
$(1,2,3,4,5,6,7,7,9,8,8,8,10,10,10,11,11,11)=(1,2,3,4,5,6,7,7,9)\circ(8,8,8)\circ(10,10,10)\circ(11,11,11)$.
We also note that $ic(J^{*}_{n}(1))\geq ic(J^{*}_{m}(1))$ if $n>m$. The Fisher
algorithm ensures that this will always be the case.

The following theorem establishes the existence and certain characteristics of
the identity curling subsequences for a Jaco graph.

######  Theorem 3.3.

A finite Jaco graph $J^{*}_{n}(1),~{}n\geq 5$ has at least one identity
curling subsequence and all identity curling subsequences have cardinality
$3$.

###### Proof.

Illustration 3.2 shows that the result holds for $J^{*}_{n}(1),5\leq n\leq
26$. It is important to note that besides the possibility of a Jaconian set of
cardinality $3$ (see for example $J^{*}_{22}(1)$) all other vertex degree
repeats following are a double repeat. Hence, only after re-arrangement does a
vertex degree preceding the Jaconian vertex degree, and equal to the doubly
repeated vertex degree cluster to provide a $3$-entry identity curling
subsequence. Assume that it holds for $J^{*}_{k}(1),k>26$, which we know has
at least $4$ such distinct double vertex degree repeats. Also assume
$J^{*}_{k}(1)$ has $l\geq 4$ such distinct double vertex degree repeats
following the Jaconian set. Consider any such double vertex degree repeat at
say $v_{j},v_{j+1}$.

Now, consider $J_{k+1}(1)$. From the definition of a Jaco graph the edges
$v_{j}v_{k+1}$ and $v_{j+1}v_{k+1}$ with perhaps $d(v_{j})=j$. Since, the
vertex degrees of all vertices $v_{h>j+1}$ are less than $d(v_{j+1})$ no
triple vertex degree repeat is possible. This holds for all $l\geq 4$ double
degree repeats. Hence, $J_{k+1}(1)$ has at least $l\geq 4$ double degree
repeats. By re-arranging the degree sequence, subsequences of triple vertex
degree repeats can be clustered. This completes the proof. ∎

######  Corollary 3.4.

For a Jaco graph $J_{n}(1),n\geq 1$ we have $1\leq cn(J^{*}_{n}(1))\leq 3$.

###### Proof.

Illustration 3.2 shows that $cn(J^{*}_{1}(1))=1$,
$cn(J^{*}_{2}(1))=cn(J^{*}_{3}(1))=cn(J^{*}_{4}(1))\\\ =2$ and from Theorem
3.3, it follows that $cn(J^{*}_{n}(1))=3,n\geq 5$. ∎

##  4 Curling Number of Set-Graphs

In this section, we discuss the curling number of a special class of graphs
called set-graphs. Let us first recall the definition of set-graphs as given
in [9].

######  Definition 4.1.

[9] Let $A^{(n)}=\\{a_{1},a_{2},a_{3},\ldots,a_{n}\\},n\in\mathbb{N}$ be a
non-empty set and the $i$-th $s$-element subset of $A^{(n)}$ be denoted by
$A_{s,i}^{(n)}$. Now consider
$\mathcal{S}=\\{A_{s,i}^{(n)}:A_{s,i}^{(n)}\subseteq
A^{(n)},A_{s,i}^{(n)}\neq\emptyset\\}$. The set-graph corresponding to set
$A^{(n)}$, denoted $G_{A^{(n)}}$, is defined to be the graph with
$V(G_{A^{(n)}})=\\{v_{s,i}:A_{s,i}^{(n)}\in\mathcal{S}\\}$ and
$E(G_{A^{(n)}})=\\{v_{s,i}v_{t,j}:~{}A_{s,i}^{(n)}\cap
A_{t,j}^{(n)}\neq\emptyset\\}$, where $s\neq t~{}\text{or}~{}i\neq j$.

######  Proposition 4.2.

[9] If $G$ is a set-graph, then $G$ has odd number of vertices.

An important result, which is proved in [9], on set-graphs that is relevant in
this context is given below.

######  Theorem 4.3.

[9] The vertices $v_{s,i},v_{s,j}$ of $G_{A^{(n)}}$, corresponding to subsets
$A^{(n)}_{s,i}$ and $A^{(n)}_{s,j}$ in $\mathcal{S}$ of equal cardinality, of
the set-graph $G_{A^{(n)}}$ have the same degree in $G_{A^{(n)}}$. That is,
$d_{G_{A^{(n)}}}(v_{s,i})=d_{G_{A^{(n)}}}(v_{s,j})$.

Invoking these concepts of set-graphs, let us now determine the curling number
and compound curling number of set-graphs.

######  Theorem 4.4.

Let $G=G_{A^{(n)}}$ be a set-graph with respect to a non-empty finite set
$A=A^{(n)}$. Then, the curling number of $G$ is
$\binom{n}{\lfloor\frac{n}{2}\rfloor}$ and the compound curling number of $G$
is $\prod\limits_{i=1}^{n}\binom{n}{i}$.

###### Proof.

Given that $G=G_{A^{(n)}}$ be a set-graph with respect to a non-empty finite
set $A=A^{(n)}$. By Theorem 4.3, the vertices of $G$ corresponding to the
subsets of $A$ with same cardinality, have the same degree. Let $r_{i}$ be the
degree of all vertices of $G$ corresponding to $i$-element subsets of $A$,
where $1\leq i\leq n$. We know that there are $\binom{n}{i}$ vertices in $G$
which have the same degree $r_{i}$. Then, the degree sequence of $G$ can be
written in the string form as
$(r_{1})^{\binom{n}{1}}\circ(r_{2})^{\binom{n}{2}}\circ(r_{3})^{\binom{n}{3}}\circ\ldots\circ(r_{n})^{\binom{n}{n}}$.
We know that the maximum value among the binomial coefficients
$\binom{n}{i};~{}1\leq i\leq n$ is $\binom{n}{\frac{n}{2}}$ if $n$ is even and
$\binom{n}{\frac{n-1}{2}}$ (or $\binom{n}{\frac{n+1}{2}}$). Therefore, the
curling number of $G$ is $\binom{n}{\lfloor\frac{n}{2}\rfloor}$.

Also, the compound curling number of the set-graph $G$ is
$\binom{n}{1}\binom{n}{2}\binom{n}{3}\ldots\binom{n}{n}=\prod\limits_{i=1}^{n}\binom{n}{i}$.
This completes the proof. ∎

It is quite interesting to check the number theoretic properties of the
curling number and compound curling number of set-graphs. The compound curling
number of a set-graph can be rewritten in terms of factorial and
hyperfactorial functions as follows.

######  Proposition 4.5.

The compound curling number of a set-graph $G=G_{A^{(n)}}$ is given by
$cn^{c}(G)=\frac{1}{(n\,!)^{(n+1)}}(\prod\limits_{i=1}^{n}i^{i})^{2}-1$.

###### Proof.

The hyperfactorial function $H(n)$ is defined as
$H(n)=(\prod\limits_{i=1}^{n}i^{i})$. Moreover, we have
$\prod_{i=0}^{n}\binom{n}{i}=\frac{H^{2}(n)}{\frac{1}{(n\,!)^{(n+1)}}}$ (see
[1]). Therefore, we have the compound curling number $cn^{c}(G)$ is
$\frac{1}{(n\,!)^{(n+1)}}(\prod\limits_{i=1}^{n}i^{i})^{2}-1$. ∎

######  Theorem 4.6.

The compound curling number of a set -graph $G=G_{A^{(n)}}$ is a perfect
square if and only if $n$ is an odd integer.

###### Proof.

If $n$ is an odd integer, then, by the result $\binom{n}{r}=\binom{n}{n-r}$,
we have
$cn^{c}(G)=\binom{n}{1}\binom{n}{2}\ldots\binom{n}{\frac{n-1}{2}}\binom{n}{\frac{n+1}{2}}\ldots\binom{n}{n-1}\binom{n}{n}=\binom{n}{1}\binom{n}{2}\ldots\binom{n}{\frac{n-1}{2}}\binom{n}{\frac{n-1}{2}}\ldots\binom{n}{1}.1=(\prod\limits_{i=1}^{n}\binom{n}{\frac{n-1}{2}})^{2}$.

If $n$ is an even integer, then by using the result, we have
$cn^{c}(G)=(\prod\limits_{i=1}^{n}\binom{n}{\frac{n}{2}-1})^{2}\binom{n}{\frac{n}{2}}$.
Since, $\binom{n}{\frac{n}{2}}$ is not a perfect square for any positive
integer $n$, $cn^{c}(G)$ can not be a perfect square in this case. This
completes the proof. ∎

##  5 Conclusion

In this paper, we have introduced the concepts of curling number and compound
curling number of a given graph and discussed certain properties of these new
parameters for certain standard graphs and digraphs. More problems regarding
the curling number and compound curling number of certain other graph classes,
graph operations, graph products and graph powers are still to be settled.
Some other problems we have identified in this area for further works are the
following.

######  Problem 5.1.

Determine the condition for the compound curling number of a spanning subgraph
of a given regular graph $G$ is greater than that of $G$.

######  Problem 5.2.

Characterise the graphs in accordance with its compound curling number and
that of its spanning subgraphs.

######  Problem 5.3.

Verify the existence of non-regular graphs whose curling numbers and compound
curling numbers are equal and characterise these graphs if exists.

There are more problems in this area which seem to be promising for further
investigations. All these facts highlights a wide scope for further studies in
this area.

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