Comparative studies demonstrate that bacteriophages have many common features on the molecular level and common principle of interaction with a bacterium cell, although components that trigger adsorption of phages to the host cell and the genome release are host dependent. Phage infection also depends on the availability of specific receptors on the cell surface, and investigation of the structure and biosynthesis of the bacterial cell membrane may be undertaken using phage-resistant mutants. Therefore there is a need to carry out further studies on phages, identifying receptors of targeted bacteria and environmental features that affect phage activity (Jończyk et al., 2011). The growing interest of the pharmaceutical and agricultural industries in phages requires more information on phage interactions, survivability and methods of their preservation. Structural studies revealed many similarities between bacteriophages and animal cell viruses. The chances of success in using bacteriophages as model systems for animal cell viruses and eventually as medical therapy are much better given our current extensive knowledge of bacteriophage biology following the advances in their molecular structural biology.

#### **7. Acknowledgments**

The author is grateful to Dr Helen White for the helpful comments during the preparation of this manuscript. EVO was also supported by BBSRC grant BB/F012705/1

#### **8. References**

22 Bacteriophages

refined to a 4.5 Å resolution (Figure 6, Jiang et al., 2008). The quality of the map allowed tracing the backbone chain of gp7. Comparison of the models has shown local discrepancies between subunits at the N- terminus and the E-loop in different subunits of gp7 within the hexamers of the capsid. Interestingly, a connection between E-loops of neighbouring subunits possibly exists, but the resolution was not sufficient to reveal it. Moreover, additional density was located between the gp7 monomers. This density has been assigned to the gp10 decoration protein that consists mainly of beta-sheets and two short alphahelices. A back-to-back dimer of gp10 is positioned at the two-fold axes and makes contact with six gp7 subunits through the N-termini and the E-loops. It was suggested that gp10 'staples' the underlying gp7 capsomeres to cement the gp5 cage so that it withstands the

The Epsilon15 capsid volume can accommodate up to 90kb dsDNA. Since the Epsilon15 genome is only ~40kb, there is ample space for a protein core of this size in the capsid chamber. The core has a cylindrical shape with a length of ~200 Å and diameter of ~180 Å. The protein core may facilitate the topological ordering of the dsDNA genome during packaging and/or release as suggested for T7 core. At the virion's tail vertex, six tails pikes attach to a central 6-fold-symmetric tail hub of the length ~170 Å. This hub may be equivalent to *Salmonella typhimurium* bacteriophage P22's hub. The hub is connected to the portal ring inside the capsid. The Epsilon 15 genome winds around the core, with a short segment of terminal DNA passing through the axis of the core and portal (Jiang et al., 2006)

Bacteriophages represent an example of amazing molecular machines with powerful motors energised by ATP hydrolysis and puncturing devices allowing to inject viral genome into the host cells. As more and more phage structures been studied a general theme emerges pointing to a common bacteriophage ancestor from which they all inherited essentially the same capsid protein fold and other elements of their organisation: capsids, tails, portal complexes, tail fibres, and other components. The number of phages that were discovered, purified, and studied by biochemical, and biophysical methods increased tremendously during the last decade. New technologies used for their studies both on the microbiological and molecular levels made it possible to analyse their evolutionary relationship and origins of the host range specificity. One of the powerful techniques in the structural biology of phages is the modern cryo-EM that recently allowed to reach close to atomic resolution level of details in the EM reconstructions (Hryc et al., 2011; Zhou, 2011; Grigorieff & Harrison, 2011). Understanding of the mechanisms which determine the host-range is required to solve many practical questions related to infectious human and animal diseases caused by bacteria, and quality food and its production (e.g. dairy products). A study conducted in Japan has demonstrated the efficiency of phages against bacterial infections of cultured fish (Nakai & Park, 2002). The use of bacteriophage as antimicrobial agents is based on the lytic phages that kill bacteria via lysis, which destroys the bacterium and makes its adaptation nearly impossible. High bacteriophage resistance for external factors is important for the stability of phage preparations. However, this stability is disadvantageous for industry

Comparative studies demonstrate that bacteriophages have many common features on the molecular level and common principle of interaction with a bacterium cell, although

pressure from packed dsDNA (Jiang et al., 2008).

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**2**

*1,2México 3Colombia*

**Some Reflections on the Origin**

Francisco Martínez-Pérez3 and Rosa Ma. Bermúdez1

*2Facultad de Química, Universidad Nacional Autónoma de México D. F. 3Laboratorio de Microbiología y Mutagénesis Ambiental, Escuela de Biología,*

*Centro de Investigación y de Estudios Avanzados del IPN,*

Luis Kameyama1, Eva Martínez-Peñafiel1, Omar Sepúlveda-Robles1,

Monophyletic theory of the origin of life postulates that all cellular organisms have evolved from a common ancestor. This is based on nucleotide sequence analyses of rRNA genes, which all cellular organisms possess (Woese et al., 1975). On this basis and supported by other observations, the first common ancestor dates back to ~ 3.7 billion years. However, the scenario is not equivalent for the viruses, since they lack these genetic elements. In fact, there is such substantial diversity in viral genome structures (dsDNA, dsRNA, ssDNA, ssRNA) that it has proven extremely difficult to answer several key evolutionary questions. Do they co-evolve with their hosts? How do viruses first infect a new species? When was the first virus? Thus, a polyphyletic origin of viruses has been proposed (Bamford, 2003). It has been proposed that this could have been even before the appearance of the first cell. Although Boyer et al. (2010) have suggested that the eukaryote viruses may have appeared just after or simultaneously with the emergence of modern eukaryote lineages. However, there are other proposals which state that as new species appeared (of any of these three domains: Eukarya, Archaea and Bacteria), and after a certain period of time, their new infective viruses emerged. Nevertheless, information about common ancestor(s) related to viruses is still an enigma. Similarly, if we ask the same question for bacteriophages, or simply, when did the first lambdoid phage emerge? The answer is also unknown. Even with the support of bioinformatics and phage genomic knowledge, and quite possibly due to the lack of specific strategies and/or design methodologies, and phage genomic complexity the

In this chapter, in an attempt to address this issue in *grosso modo*, we propose the analysis of two genomic regions of lambdoid phages, regions that are dissimilar regarding their nucleotide variability and stability: (1) The variable and not essential region that confers immunity to the lysogenic bacteria against phage superinfection counterparts, and is related

**1. Introduction**

problem has not yet been resolved.

**of Lambdoid Bacteriophages**

Zaira Y. Flores-López1, Leonor Fernández2,

*1Departamento de Genética y Biología Molecular,*

*Universidad Industrial de Santander, Bucaramanga,*


**8. Conclusion**

Mid-GI bleeding is common in our day-to-day clinical practice. Capsule endoscopy and imaging studies have made the diagnostic evaluations much easier than before. Balloon-assisted enteroscopy and spiral enteroscopy are generally done for therapeutic interventions. Interventional radiology and surgery are required if there is massive bleeding or endoscopic therapeutic interventions fail. After hemostasis is obtained, treatment of the underlying condition should be done. Patient's age, comorbidities, pros and cons of the procedures, and radiological and surgical interventions should be considered.

*Digestive System - Recent Advances*


**8. Conclusion**

In textile industry, the surface modification of fibers is correlated to surface properties such as water repellence and adhesion. In the recent years, there have been many researches in developing surface modification methods to improve the surface properties of textile materials in order to develop new market products. Several surface modification methods are employed. The irradiation treatments have been ahead of recognition as a surface modification technique. Much attention has been paid to treating the surfaces of materials with irradiation methods in the past few decades. In general, treating the surface of a material with irradiation methods can cause physical and chemical changes to occur at the surface but affect the bulk properties of the material little.


**2. Variability among individuals: why it is important to study gene polymorphisms**

Gene polymorphisms are the most common type of genetic variations in humans. They are present in the human population at frequency higher than 1% and differ from DNA mutations which are generally observed at extremely low frequency and in a restricted number of individuals. While gene polymorphisms are not necessarily associated to a specific disease, the gene mutation is generally known to cause a genetic disease. In humans the simplest type of polymorphism is the single nucleotide polymorphisms (SNPs), representing the most common type of nucleotide variation where a single base is substituted by another one (**Figure 1**). A SNP in the coding region of a gene may have four different effects on the resulting protein: (1) *synonymous substitution*, also known as silent mutation, causing no amino acid change in the protein, (2) *non-synonymous substitution* where a nucleotide mutation alters the amino acid sequence of a protein, (3) *missense substitution* consisting of an amino acid change with another and (4) *nonsense substitution* resulting in end of protein translation by a termination codon. About half of all the coding sequences of SNPs end up in non-synonymous codon changes. However SNPs may occur also in the regulatory region of the gene, affecting various properties of the protein. These SNPs may influence the protein in terms of quantity and quality, activity, processing and trafficking [2]. In the last 20 years, SNPs have raised a lot of

**67**

*The Role of Genetic Polymorphisms in the Occupational Exposure*

interest in many scientific fields related to public health and disease. They are also investigated in many scientific research areas, ranging from the human ethnicity study to the genetics of populations, personalized medicine, pathology, epidemiology, pharmacology, immunology and nutrition. More recently they have been considered relevant also for their role in the exposure science, a novel scientific branch linked to toxicology which characterizes and elucidates the contact of humans with dangerous chemical, physical and biological agents which represent a potential risk for the human health [3]. Actually the environmental exposure, in which also the occupational exposure is comprised, is acknowledged to play an important role in common chronic diseases, representing a major health concern in the economically developed countries. In this context the study of SNPs has a fundamental role to detect the human response to toxic and dangerous substances. Although the majority of SNPs of the human genome are of low prevalence [4], including the genes implicated in metabolism of environmental chemicals, these may substantially contribute to increase the population disease burden [5]. Since the exposure factors are numerous and several chronic diseases remain uncertain, it is fundamental to study the human genetic variation to understand the exposure-disease associations within the global population [6]. On the whole the genetic diversity associated to the variability of polymorphisms and to the genetic recombination is a valuable resource for humans and other living organisms. The interactions between the genetics of human beings and the surrounding environment reciprocally shape one with each other in order to reach an equilibrium. However such equilibrium might be perturbed by several exogenous factors such as the influence of natural selection, the global climate change, the re-emergence of extinct diseases and the admixing of different ethnicities due to massive migrations from one country to another. All these factors may contribute to change the variation of the gene polymorphism pool in the worldwide population. Nowadays the characterization of gene polymorphisms of the human population is carried out on individuals belonging to the same ethnicity so as to identify ethnic-specific gene/allele frequencies. This strategy should help geneticists to identify the genotype frequencies typical of members of a community sharing the same language, culture, religion, tradition, nationality, ancestry, nutrition, habits and lifestyle which influence the genetic background of the group [7]. For this reason the investigational studies of polymorphic genes refer more to the concept of ethnicity than to the race. The genetic characterization of separate ethnic groups provides useful information to evaluate the difference in the susceptibility risk of each cluster. Several papers have been published to report the typical genotype frequency of different gene polymorphisms in selected ethnicities [8–11]. However, due to the abundance of variation in gene polymorphisms, the characterization of the gene pool of a specific group of individuals is incomplete, and the genotype frequencies are known only for polymorphisms which have been investigated for their role or association to a specific susceptibility or disease. To our knowledge the commonest public databases collecting the gene polymorphisms available to the scientific community are the following: (1) the single nucleotide polymorphism database (dbSNP), a public archive for genetic variation hosted by the National Center for Biotechnology Information (NCBI) (https://www.ncbi.nlm. nih.gov/snp), and (2) the Ensembl project of genome databases for vertebrates and other eukaryotic species (http://grch37.ensembl.org/Homo\_sapiens/Variation), which is the one we are using in the study of the human SNPs. These two archives represent a fundamental resource for our investigational studies on human genetic diversity when the laboratory genotyping is not feasible. In this chapter we describe some of our studies carried out during the biomonitoring campaigns of exposed workers where the biomarkers of dose, effect and susceptibility have been assessed in the occupational exposure. In addition we show also a statistical model we

*DOI: http://dx.doi.org/10.5772/intechopen.86975*

**Figure 1.** *Variation of single nucleotide in the human DNA coding sequence among individuals.*

## *The Role of Genetic Polymorphisms in the Occupational Exposure DOI: http://dx.doi.org/10.5772/intechopen.86975*

*The Recent Topics in Genetic Polymorphisms*

**polymorphisms**

increase further the genetic variation in the population.

of the surrounding environment on our genotype and on our physiological, physical and behavioral habits. Such influence, namely, epigenetics, means that the heritable changes in gene expression are not due to alteration in the DNA sequence but to some modifications occurring upon the DNA without changing the genetic code. That said, although we are aware the genetic variability is something peculiar to every human being, regardless of the inheritance of favorable or unfavorable genes, the transmission to the offspring will be modulated by epigenetics contributing to

**2. Variability among individuals: why it is important to study gene**

Gene polymorphisms are the most common type of genetic variations in humans. They are present in the human population at frequency higher than 1% and differ from DNA mutations which are generally observed at extremely low frequency and in a restricted number of individuals. While gene polymorphisms are not necessarily associated to a specific disease, the gene mutation is generally known to cause a genetic disease. In humans the simplest type of polymorphism is the single nucleotide polymorphisms (SNPs), representing the most common type of nucleotide variation where a single base is substituted by another one (**Figure 1**). A SNP in the coding region of a gene may have four different effects on the resulting protein: (1) *synonymous substitution*, also known as silent mutation, causing no amino acid change in the protein, (2) *non-synonymous substitution* where a nucleotide mutation alters the amino acid sequence of a protein, (3) *missense substitution* consisting of an amino acid change with another and (4) *nonsense substitution* resulting in end of protein translation by a termination codon. About half of all the coding sequences of SNPs end up in non-synonymous codon changes. However SNPs may occur also in the regulatory region of the gene, affecting various properties of the protein. These SNPs may influence the protein in terms of quantity and quality, activity, processing and trafficking [2]. In the last 20 years, SNPs have raised a lot of

**66**

**Figure 1.**

*Variation of single nucleotide in the human DNA coding sequence among individuals.*

interest in many scientific fields related to public health and disease. They are also investigated in many scientific research areas, ranging from the human ethnicity study to the genetics of populations, personalized medicine, pathology, epidemiology, pharmacology, immunology and nutrition. More recently they have been considered relevant also for their role in the exposure science, a novel scientific branch linked to toxicology which characterizes and elucidates the contact of humans with dangerous chemical, physical and biological agents which represent a potential risk for the human health [3]. Actually the environmental exposure, in which also the occupational exposure is comprised, is acknowledged to play an important role in common chronic diseases, representing a major health concern in the economically developed countries. In this context the study of SNPs has a fundamental role to detect the human response to toxic and dangerous substances. Although the majority of SNPs of the human genome are of low prevalence [4], including the genes implicated in metabolism of environmental chemicals, these may substantially contribute to increase the population disease burden [5]. Since the exposure factors are numerous and several chronic diseases remain uncertain, it is fundamental to study the human genetic variation to understand the exposure-disease associations within the global population [6]. On the whole the genetic diversity associated to the variability of polymorphisms and to the genetic recombination is a valuable resource for humans and other living organisms. The interactions between the genetics of human beings and the surrounding environment reciprocally shape one with each other in order to reach an equilibrium. However such equilibrium might be perturbed by several exogenous factors such as the influence of natural selection, the global climate change, the re-emergence of extinct diseases and the admixing of different ethnicities due to massive migrations from one country to another. All these factors may contribute to change the variation of the gene polymorphism pool in the worldwide population. Nowadays the characterization of gene polymorphisms of the human population is carried out on individuals belonging to the same ethnicity so as to identify ethnic-specific gene/allele frequencies. This strategy should help geneticists to identify the genotype frequencies typical of members of a community sharing the same language, culture, religion, tradition, nationality, ancestry, nutrition, habits and lifestyle which influence the genetic background of the group [7]. For this reason the investigational studies of polymorphic genes refer more to the concept of ethnicity than to the race. The genetic characterization of separate ethnic groups provides useful information to evaluate the difference in the susceptibility risk of each cluster. Several papers have been published to report the typical genotype frequency of different gene polymorphisms in selected ethnicities [8–11]. However, due to the abundance of variation in gene polymorphisms, the characterization of the gene pool of a specific group of individuals is incomplete, and the genotype frequencies are known only for polymorphisms which have been investigated for their role or association to a specific susceptibility or disease. To our knowledge the commonest public databases collecting the gene polymorphisms available to the scientific community are the following: (1) the single nucleotide polymorphism database (dbSNP), a public archive for genetic variation hosted by the National Center for Biotechnology Information (NCBI) (https://www.ncbi.nlm. nih.gov/snp), and (2) the Ensembl project of genome databases for vertebrates and other eukaryotic species (http://grch37.ensembl.org/Homo\_sapiens/Variation), which is the one we are using in the study of the human SNPs. These two archives represent a fundamental resource for our investigational studies on human genetic diversity when the laboratory genotyping is not feasible. In this chapter we describe some of our studies carried out during the biomonitoring campaigns of exposed workers where the biomarkers of dose, effect and susceptibility have been assessed in the occupational exposure. In addition we show also a statistical model we

previously elaborated to identify ethnic-specific differences in the susceptibility risk to the typical exposure found in the workplace. In such model a statistical analysis has been done using the publicly available genotype frequency of four ethnic groups (Africans, East Asians, South Asians and Europeans) downloaded from the Ensembl project of genome databases.


**4. The Crossing Number of** *G∗* **+** *Dn*

A drawing *D* of *G*∗ + *Dn* is said to be antipode-free if cr*D*(*Ti* , *Tj* ) ≥ 1 for any two different vertices *ti* and *tj*. In the proof of Theorem 2, the following statements related to some restricted subdrawings of the graph *G*∗ + *Dn* are required.

**Lemma 1.** *Let D be a good and antipode-free drawing of G*∗ + *Dn, n* > 1*, with the vertex notation of the graph G*∗ *in such a way as shown in Figure 1a. For Ti* ∈ *RD, let* A*k* ∈ M*D be a configuration of the corresponding subgraph Fi* = *G*∗ ∪ *Ti for some k* ∈ {1, ... , 6}*. If there is a Tj* ∈ *SD such that* cr*D*(*Ti* , *Tj* ) = 1*, then all possible* rot*D*(*tj*) *are given in Table 2.*

**Table 2.** The corresponding rotations of *tj*, for *Ti* ∈ *RD*, *Fi* = *G*∗ ∪ *Ti* and *Tj* ∈ *SD* satisfying the restriction cr*D*(*Ti* , *Tj* ) = 1.

**Proof.** Assume the configuration A1 of the subgraph *Fi* = *G*∗ ∪ *Ti* for some *Ti* ∈ *RD*, i.e., rot*D*(*ti*)=(13245). The subdrawing of *Fi* induced by *D* contains just five regions with *ti* on their boundaries, see Figure 2. If there is a *Tj* ∈ *SD* such that cr*D*(*Ti* , *Tj* ) = 1, then the vertex *tj* must be placed in the region with the four vertices *v*1, *v*2, *v*4, and *v*5 of *G*∗ on its boundary. Besides that only the edge *v*1*v*2 of *G*∗ can be crossed by *tjv*3, and cr*D*(*Ti* , *Tj* ) = 1 is fulfilling for *Tj* with rot*D*(*tj*)=(14523) if *tjv*4 crosses *tiv*5. The same idea also force that the rotations of the vertex *tj* are (15423), (12345), (12354), (12453), and (12543) for the remaining configurations A2, A3, A4, A5, and A6 of *Fi* , respectively.

**Corollary 1.** *Let D be a good and antipode-free drawing of G*∗ + *Dn, for n* > 3*, with the vertex notation of the graph G*∗ *in such a way as shown in Figure 1a. If Ti* , *Tj , and Tk* ∈ *RD are three different subgraphs with* cr*D*(*Ti* , *Tj* ) = 1*,* cr*D*(*Ti* , *Tk*) = 1 *and such that Fi , Fj , and Fk have three mutually different configurations from any of the sets* {A1, A4, A6}*,* {A2, A3, A5}*,* {A3, A2, A6}*,* {A1, A4, A5}*,* {A2, A4, A5}*, and* {A1, A3, A6}*, then*

$$\mathsf{crr}\_{D}(T^{i}\cup T^{j}\cup T^{k},T^{l})\geq\mathsf{6} \qquad\qquad\qquad\qquad\text{for any }T^{l}\in\mathbb{S}\_{D}.$$

*i.e.,*

$$\operatorname{crr}\_D(G^\* \cup T^i \cup T^j \cup T^k, T^l) \ge \mathcal{T} \qquad\qquad\qquad\text{for any } T^l \in \mathcal{S}\_D.$$

**Proof.** Let us assume the configurations A1 of *Fi* , A4 of *Fj* , and A6 of *Fk* with respect to the restrictions cr*D*(*Ti* , *Tj* ) = cr*D*(*Ti* , *Tk*) = 1 and recall that they are represented by the cyclic permutations rot*D*(*ti*)=(13245), rot*D*(*tj*)=(15432), and rot*D*(*tk*)=(13542). If there is a subgraph *Tl* ∈ *SD* with cr*D*(*Ti* , *Tl* ) = 1, then the subgraph *Fl* can be represented only by rot*D*(*tl*)=(14523), where the edge *tlv*3 crosses *v*1*v*2 of *G*∗ and either *tlv*4 or *tlv*5 crosses corresponding edge of *Ti* . Any such subgraph *Tl* must cross edges of both subgraphs *Tj* and *Tk* at least twice because the minimum number of interchanges of adjacent elements of (14523) required to produce (15432)=(12345) and (13542)=(12453) is two. Clearly, if cr*D*(*Tj* , *Tl* ) > 2 or cr*D*(*Tk*, *Tl* ) > 2, we obtain the desired result cr*D*(*Ti* ∪ *Tj* ∪ *Tk*, *Tl* ) ≥ 1 + 3 + 2 = 6. Further, if cr*D*(*Tj* , *Tl* ) = 2 and cr*D*(*Tk*, *Tl* ) = 2, then the edge *tiv*5 is crossed by *tlv*4 in *D*(*Ti* ∪ *Tj* ∪ *Tl* ) and also *tiv*4 by *tlv*5 in *D*(*Ti* ∪ *Tk* ∪ *Tl* ), respectively. However, then cr*D*(*Ti* , *Tl* ) ≥ 2, which contradicts the fact that cr*D*(*Ti* , *Tl* ) = 1 in *D*(*Ti* ∪ *Tj* ∪ *Tk* ∪ *Tl* ).

If there is a *Tl* ∈ *SD* with cr*D*(*Tj* , *Tl* ) = 1, then the subgraph *Fl* is represented only by the cyclic permutation (12354). Using same properties as in the previous subcase, we have cr*D*(*Ti* , *Tl* ) ≥ 2 and cr*D*(*Tk*, *Tl* ) ≥ 3. This in turn implies that cr*D*(*Ti* ∪ *Tj* ∪ *Tk*, *Tl* ) ≥ 2 + 1 + 3 = 6. Of course, we can apply the same idea for the case of cr*D*(*Tk*, *Tl* ) = 1.

To finish the proof, let us consider a subgraph *Tl* ∈ *SD* with cr*D*(*Ti* , *Tl* ) = 2, cr*D*(*Tj* , *Tl* ) = 2, and cr*D*(*Tk*, *Tl* ) = 2. This enforces that the minimum number of interchanges of adjacent elements of rot*D*(*tl*) required to produce (13245)=(15423), (15432)=(12345), and (13542)=(12453) must be exactly two. However, it is not difficult to show that such cyclic permutation does not exist. Similar arguments can be applied for remaining five cases (or using the transformations Π1 and Π2), and the proof is complete.

**Corollary 2.** *Let D be a good and antipode-free drawing of G*∗ + *Dn, for n* > 3*, with the vertex notation of the graph G*∗ *in such a way as shown in Figure 1a. If Ti* , *Tj , and Tk* ∈ *RD are three different subgraphs such that Fi , Fj , and Fk have three mutually different configurations from any of the sets* {A1, A3, A5} *and* {A2, A4, A6}*, then*

$$\mathsf{crr}\_{D}(T^{i}\cup T^{j}\cup T^{k},T^{l}) \geq 5\tag{10.43} \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad for \ any \ T^{l} \in \mathcal{S}\_{D,l}$$

*i.e.,*

$$\operatorname{crr}\_D(G^\* \cup T^i \cup T^j \cup T^k, T^l) \ge 6 \qquad \qquad \qquad \text{for any } T^l \in S\_D.$$

**Proof.** Let us assume the configurations A1 of *Fi* , A3 of *Fj* , and A5 of *Fk*. If there is a subgraph *Tl* ∈ *SD* with cr*D*(*Ti* , *Tl* ) = 1, then the subgraph *Fl* can be represented only by the cyclic permutations (14523). Uniqueness of all rotations in Table 2 confirms that cr*D*(*Tj* , *Tl* ) ≥ 2 and cr*D*(*Tk*, *Tl* ) ≥ 2. Hence, cr*D*(*Ti* ∪ *Tj* ∪ *Tk*, *Tl* ) ≥ 1 + 2 + 2 = 5, and the similar way can be applied for the case if cr*D*(*Tj* , *Tl* ) = 1 or cr*D*(*Tk*, *Tl* ) = 1 with *Tl* ∈ *SD*. It remains to consider the case where cr*D*(*Ti* , *Tl* ) ≥ 2, cr*D*(*Tj* , *Tl* ) ≥ 2, and cr*D*(*Tk*, *Tl* ) ≥ 2, which yields that cr*D*(*Ti* ∪ *Tj* ∪ *Tk*, *Tl* ) ≥ 2 + 2 + 2 = 6 clearly holds for any such *Tl* , as claimed. The proof proceeds in the similar way for the second triple of configurations {A2, A4, A6}, and this completes the proof.

**Lemma 2.** cr(*G*∗ + *D*2) = 1*.*

**Proof.** If we consider the configurations A2 of *Fi* and A3 of *Fj* , then one can easily find a subdrawing of *Ti* ∪ *Tj* in which cr*D*(*Ti* , *Tj* ) = 1, i.e., cr(*G*∗ + *D*2) ≤ 1. The graph *G*∗ + *D*2 contains a subgraph that is a subdivision of the complete graph *K*5 and it is well-known by Guy [32] that cr(*K*5) = 1. As cr(*G*∗ + *D*2) ≥ cr(*K*5) = 1, the proof of Lemma 2 is complete.

**Theorem 2.** cr(*G*∗ + *D*1) = 0 *and* cr(*G*∗ + *Dn*) = *n*2 − 2*n* + - *n* 2 . *for n* ≥ 2*, i.e.,* cr(*G*∗ + *Dn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . *for n even and* cr(*G*∗ + *Dn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . − 1 *for n odd at least 3.*

**Proof.** The graph *G*∗ + *D*1 is planar, hence cr(*G*∗ + *D*1) = 0. For *n* ≥ 2, both special drawings in Figures 4 and 5 produce *n*2 − 2*n* + - *n* 2 . crossings, and so cr(*G*∗ + *Dn*) ≤ *n*2 − 2*n* + - *n* 2 . . The opposite inequality can be proved by induction on *n*, and the result holds for *n* = 2 by Lemma 2. For some *n* ≥ 3, suppose a drawing *D* of *G*∗ + *Dn* with

$$\text{crr}\_D(G^\* + D\_n) < n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor \tag{3}$$

and that

$$\text{crr}(G^\* + D\_m) = m^2 - 2m + \left\lfloor \frac{m}{2} \right\rfloor \quad \text{for any integer } 2 \le m < n. \tag{4}$$

**Figure 4.** The good drawing of *G*∗ + *Dn* with 4- *n* 2 .- *n*−1 2 . + - *n* 2 . crossings for *n* even, *n* ≥ 2.

**Figure 5.** The good drawing of *G*∗ + *Dn* with 4- *n* 2 .- *n*−1 2 . + - *n* 2 . − 1 crossings for *n* odd, *n* ≥ 3, where three subgraphs *T*1, *T*2, and *Tn* are fixed.

Let us first show that *D* must be antipode-free. Suppose that, without loss of generality, cr*D*(*Tn*−1, *Tn*) = 0. If at least one of *Tn*−1 and *Tn*, say *Tn*, does not cross *G*∗, it is not difficult to verify in Figure 1 that {*Tn*−1, *Tn*} ⊆ *RD*, i.e., cr*D*(*G*∗, *Tn*−1 ∪ *Tn*) ≥ 1. By (1), we already know that cr*D*(*K*5,3) ≥ 4, which yields that edges of the subgraph *Tn*−1 ∪ *Tn* must be crossed at least four times by each other *Tk*. So, by fixing the subgraph *Tn*−1 ∪ *Tn* in *D*, we have

$$\text{crr}\_D(G^\* + D\_{n-2}) + \text{crr}\_D(T^{n-1} \cup T^n) + \text{crr}\_D(K\_{5,n-2}, T^{n-1} \cup T^n) + \text{crr}\_D(G^\*, T^{n-1} \cup T^n)$$

$$\geq (n-2)^2 - 2(n-2) + \left\lfloor \frac{n-2}{2} \right\rfloor + 0 + 4(n-2) + 1 = n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor$$

The obtained crossing number contradicts the assumption (3) and confirms that the considered drawing *D* is antipode-free. For easier reading, if *r* = |*RD*| and *s* = |*SD*|, then again (3) together with cr*D*(*K*5,*n*) ≥ 4 - *n* 2 .- *n*−1 2 . using (1) imply the following inequality with respect to possible edge crossings of *G*∗ in *D*:

$$\text{crr}\_D(\mathbf{G}^\*) + s + 2(n - r - s) < \left\lfloor \frac{n}{2} \right\rfloor. \tag{5}$$

The inequality (5) forces more than / *n* 2 0 subgraphs *Ti* by which edges of *G*∗ are not crossed, that is, *r* ≥ / *n* 2 0 + 1 ≥ 3 and *s* < - *n* 2 . . Of course, if *n* is odd then previous inequalities could be strengthened, but this is not necessary in the following process of obtaining a contradiction with number of crossings in *D*. Moreover, if *n* = 3 then *r* = 3, and cr*D*(*G*∗ + *D*3) ≥ cr(*K*5,3) = 4 with the assumption (3) enforce *n* at least four.

Case 1: cr*D*(*G*∗) = 0 and choose the vertex notation of the graph *G*∗ in such a way as shown in Figure 1a. In this case, we deal with configurations from the nonempty set M*D*. As the set *RD* is nonempty, recall that

$$\sum\_{i \in I\_{\mathcal{S}}} \alpha\_i + \sum\_{i \in I\_{\mathcal{S}}} \alpha\_i = r \ge 3\dots$$

Let us first suppose that either *α*1 + *α*3 + *α*5 = 0 or *α*2 + *α*4 + *α*6 = 0. For the rest of the proof we may therefore assume that *α*2 + *α*4 + *α*6 = 0, that is, *α*1 + *α*3 + *α*5 > 0. Since G*D* is the subgraph of G induced by *VD* with respect to weights 2 of all its edges (without possible loops), three possible subcases presented in Figure 6 may occur:

**Figure 6.** Three possible components of the graph G*D* if *α*2 = *α*4 = *α*6 = 0. (**a**): *αi* > 0 for each *i* ∈ *Io*; (**b**): *αi* > 0 and *αj* > 0 for exactly two different *i*, *j* ∈ *Io*; (**c**): *αi* > 0 for only one *i* ∈ *Io*.

(a) *αi* > 0 for each *i* ∈ *Io*. Let us assume three subgraphs *Tn*−2, *Tn*−1, *Tn* ∈ *RD* such that *Fn*−2, *Fn*−1 and *Fn* have three mutually different configurations from the set M*D* = {A1, A3, A5}. Then, cr*D*(*Tn*−2 ∪ *Tn*−1 ∪ *Tn*, *Ti* ) ≥ 4 + 2 + 2 = 8 holds for any other *Ti* ∈ *RD* by summing values in corresponding three rows of Table 1, and cr*D*(*G*∗ ∪ *Tn*−2 ∪ *Tn*−1 ∪ *Tn*, *Ti* ) ≥ 6 is true by Corollary 2 for any *Ti* ∈ *SD*. Then, by fixing the graph *G*∗ ∪ *Tn*−2 ∪ *Tn*−1 ∪ *Tn*

$$\text{crr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor + 8(r-3) + 6s + 7(n-r-s) + 6s$$

$$= 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor + 7n + r - s - 18 \ge 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor$$

$$+ 7n + \left( \left\lceil \frac{n}{2} \right\rceil + 1 \right) + \left( 1 - \left\lfloor \frac{n}{2} \right\rfloor \right) - 18 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

(b) Assuming that *αi* > 0 for exactly two *i* ∈ *Io*, without lost of generality, let us consider two different subgraphs *Tn*−1, *Tn* ∈ *RD* such that *Fn*−1 and *Fn* have configurations A1 and A3, respectively. As M*D* = {A1, A3}, we have cr*D*(*Tn*−1 ∪ *Tn*, *Ti* ) ≥ 4 + 2 = 6 for any *Ti* ∈ *RD*, *i* = *n* − 1, *n*. Therewith, the antipode-free property of *D* forces that, cr*D*(*Tn*−1 ∪ *Tn*, *Ti* ) ≥ 2 trivially holds for any subgraph *Ti* with *i* = *n* − 1, *n*. Hence, by fixing the graph *G*∗ ∪ *Tn*−1 ∪ *Tn*

$$\operatorname{cr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 6(r-2) + 3s + 4(n - r - s) + 2$$

$$= 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 4n + 2r - s - 10 \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor$$

$$+ 4n + 2\left( \left\lceil \frac{n}{2} \right\rceil + 1 \right) + \left( 1 - \left\lfloor \frac{n}{2} \right\rfloor \right) - 10 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

(c) *αi* > 0 for only one *i* ∈ *Io*. As M*D* = {A*i*}, in the rest of the paper, we may consider *Tn* ∈ *RD* with the configuration A1 of *Fn*. Then edges of each other subgraph *Tj* ∈ *RD* cross at least four times edges of *Tn* provided by rot*D*(*tn*) = rot*D*(*tj*). Thus, by fixing the graph *G*∗ ∪ *Tn*

$$\begin{split} \operatorname{cr}\_D(G^\* + D\_n) &\geq 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 4(r-1) + 2s + 3(n - r - s) + 0 \\ &= 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 3n + r - s - 4 \geq 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor \\ &+ 3n + \left( \left\lceil \frac{n}{2} \right\rceil + 1 \right) + \left( 1 - \left\lfloor \frac{n}{2} \right\rfloor \right) - 4 \geq n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor. \end{split}$$

All three subcases contradict the assumption (3). In addition, let us suppose that *α*1 + *α*3 + *α*5 > 0 and *α*2 + *α*4 + *α*6 > 0. Remark that the subgraph G*D* can be either connected (consisting of a single component) or also disconnected with several components. Now, we are able to discuss over remaining possible components of G*D* in the following subcases:

1. There are no two adjacent edges with weights 1 in the subgraph G*D*, that is, there are four possibilities presented in Figure 7.

**Figure 7.** Four possible components of the subgraph G*D* in which there are no two adjacent edges with weights 1. Green, blue, brown, and black correspond to the values 1, 2, 3, and 4, respectively. (**a**): the complete graph *K*3 with edge weights 1, 2, and 3; (**b**): the complete graph *K*4 with edge weights 1, 1, 2, 2, 3, and 3; (**c**): the complete graph *K*2 with edge weight 1; (**d**): the complete graph *K*2 with edge weight 3.

• *wD*(*aiaj*) = 1 for some *i* ∈ *Io*, *j* ∈ *Ie*, i.e., there are three cases mentioned in Figure 7a–c. Let us consider two subgraphs *Tn*−1, *Tn* ∈ *RD* such that *Fn*−1, *Fn* have different configurations from {A*i*, A*j*}, where *i*, *j* are associated indexes. Using weights of edges in the considered component of G*D*, one can easily verify that edges of the graph *Tn*−1 ∪ *Tn* are crossed at least five times by edges of any another subgraph *Tk* ∈ *RD*. Moreover, since the minimum number of interchanges of adjacent elements of rot*D*(*tn*) required to produce rot*D*(*tn*−1) is three, any subgraph *Tk* with *k* = *n* − 1, *n* crosses edges of *Tn*−1 ∪ *Tn* at least thrice. Thus, by fixing the graph *G*∗ ∪ *Tn*−1 ∪ *Tn*

$$\operatorname{cr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 5(r-2) + 4s + 5(n-r-s) + 1$$

$$= 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 5n - s - 9 \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor$$

$$+ 5n + \left(1 - \left\lfloor \frac{n}{2} \right\rfloor \right) - 9 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

• *wD*(*aiaj*) > 1 for all *i* ∈ *Io*, *j* ∈ *Ie*, i.e., there is only one case mentioned in Figure 7d. Let us again consider two subgraphs *Tn*−1, *Tn* ∈ *RD* such that *Fn*−1, *Fn* have different configurations from {A*i*, A*j*}, where *i*, *j* are associated indexes. Then, cr*D*(*Tn*−1 ∪ *Tn*, *Tk*) ≥ 7 holds by summing edge-weights 4 and 3 for any other *Tk* ∈ *RD*. Hence, by fixing the graph *G*∗ ∪ *Tn*−1 ∪ *Tn*

$$\operatorname{cr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 7(r-2) + 3s + 4(n - r - s) + 3$$

$$= 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor + 4n + 3r - s - 11 \ge 4\left\lfloor \frac{n-2}{2} \right\rfloor \left\lfloor \frac{n-3}{2} \right\rfloor$$

$$+ 4n + 3\left(\left\lceil \frac{n}{2} \right\rceil + 1\right) + 1 - \left\lfloor \frac{n}{2} \right\rfloor - 11 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

Both discussed cases again confirm a contradiction with (3) in *D*, and so, suppose that there are two adjacent edges with weights 1 in the subgraph G*D*. Further, only in the case if the number *βj* is defined, we claim that the following two properties (6) and (7) must be also fulfilled in *D*:

$$
\beta\_j + \sum\_{i \in I\_\theta} a\_i > \left\lfloor \frac{n}{2} \right\rfloor \qquad \text{for some } j \in I\_{o\prime} \tag{6}
$$

$$
\beta\_j + \sum\_{i \in I\_\varepsilon} a\_i > \left\lfloor \frac{n}{2} \right\rfloor \qquad \text{for some } j \in I\_\varepsilon. \tag{7}
$$

For a contradiction, suppose, without loss of generality, that *β*1 + *α*1 + *α*3 + *α*5 ≤ - *n* 2 . , that is, −*α*1 − *α*3 − *α*5 − *β*1 ≥ −- *n* 2 . . In this case, from the definition of *β*1, we have *α*1 > 0, *α*4 > 0, and *α*6 > 0. Thus, in the rest of the paper, let us consider three subgraphs *Tn*−2, *Tn*−1, *Tn* ∈ *RD* such that *Fn*−2, *Fn*−1, and *Fn* have configurations A1, A4, and A6, respectively. Using values in Table 1, one can easily verify that edges of the graph *Tn*−2 ∪ *Tn*−1 ∪ *Tn* are crossed at least six times and seven times by edges of any another subgraph *Ti* ∈ *RD* with the configuration A1, A3, A5 and A2, A4, A6 of *Fi* (of course, if A*k* ∈ M*D* for some *k* ∈ *Io* ∪ *Ie* in *D*), respectively. However, from Corollary 1 we get that cr*D*(*G*∗ ∪ *Tn*−2 ∪ *Tn*−1 ∪ *Tn*, *Ti* ) ≥ 7 holds for any *Ti* ∈ *SD* provided by we can also assume that cr*D*(*Tn*−2, *Tn*−1) = 1 and cr*D*(*Tn*−2, *Tn*) = 1 due to the congruence property (If rot*D*(*tx*) and rot*D*(*ty*) are two cyclic permutations of odd length, and *Q*(rot*D*(*tx*),rot*D*(*ty*)) denotes the minimum number of interchanges of adjacent elements of rot*D*(*tx*) required to produce the inverse cyclic permutation of rot*D*(*ty*), then cr*D*(*Tx*, *Ty*) = *Q*(rot*D*(*tx*),rot*D*(*ty*)) + 2*z* for some nonnegative integer *z*, for more see Woodall [31]). Hence, by fixing the graph *G*∗ ∪ *Tn*−2 ∪ *Tn*−1 ∪ *Tn*

$$\begin{split} \text{tr}\_D(G^\* + D\_n) &\geq 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor + 6(a\_1 + a\_3 + a\_5 - 1) + 7(a\_2 + a\_4 + a\_6 - 2) + 7s \\ &+ 6\beta\_1 + 7(n - r - s - \beta\_1) + 4 = 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor + 7n - a\_1 - a\_3 - a\_5 - \beta\_1 - 16 \\ &\geq 4\left\lfloor \frac{n-3}{2} \right\rfloor \left\lfloor \frac{n-4}{2} \right\rfloor + 7n - \left\lfloor \frac{n}{2} \right\rfloor - 16 \geq n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor. \end{split}$$

The obtained crossing number also contradicts the assumption (3) of *D* and confirms that both parity properties (6) and (7) must be fulfilled in *D*.

	- (a) Let the graph G*D* consist of one component in such a way as shown in Figure 8a. Without lost of generality, let us assume that *a*2, *a*3, *a*6 are vertices of the considered path on three vertices with weight 1 of both edges. In this case, it is obvious that *α*2 + *α*3 + *α*6 = *r*. Since the number *β*3 can be defined, the property (6) forces *β*3 + *α*3 > - *n* 2 . . Further, let us also assume that *Tn* ∈ *RD* with the configuration A3 of *Fn*. Then, by fixing the graph *G*∗ ∪ *Tn*

$$\begin{split} \text{cr}\_D(G^\* + D\_n) &\geq 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 4(a\_3 - 1) + 1(a\_2 + a\_6) + 4\beta\_3 + 2s \\ &+ 3(n - r - s - \beta\_3) = 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 3n + (a\_3 + \beta\_3 - a\_2 - a\_6) \\ &- (s + a\_2 + a\_6) - 4 \geq 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 3n + 0 - \left\lfloor \frac{n}{2} \right\rfloor - 4 \geq n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor. \end{split}$$

(b) Let the graph G*D* consist of one component in such a way as shown in Figure 8b. Without lost of generality, let us assume that *a*2, *a*3, *a*6 are vertices of the considered path on three vertices with weight 1 of both edges and let *a*2, *a*4, *a*6 be vertices of the 3-cycle with respect to weight 2 of all its edges. In this case, it is obvious that *α*2 + *α*3 + *α*4 + *α*6 = *r*. The property (6) enforces again *β*3 + *α*3 > - *n* 2 . because the number *β*3 can be defined. Further, if *Tn* ∈ *RD* is assumed with the configuration A4 of *Fn*, then by fixing the graph *G*∗ ∪ *Tn*

$$\text{crr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 3(\alpha\_3 + \beta\_3) + 2(n - \alpha\_3 - \beta\_3 - 1).$$

$$\ge 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 2n + \left( \left\lfloor \frac{n}{2} \right\rfloor + 1 \right) - 2 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

(c) Let the graph G*D* consist of one component in such a way as shown in Figure 8c–e. Let us take a maximal path *Pk* on *k* vertices as the subgraph of G*D* with weights 1 on all its edges. If *ai* and *aj* are two inner vertices of *Pk* with *i* + 1 ≡ *j* (mod 2) for which the numbers *βi* and *βj* satisfy the parity properties (6) and (7), then addition of both inequalities thus obtained enforces a contradiction

$$m \ge \beta\_i + \beta\_j + r \ge 2\left(\left\lfloor \frac{n}{2} \right\rfloor + 1\right).$$

The obtained contradictions in all three cases complete the proof for the planar subdrawing of *G*∗ induced by *D* given in Figure 1a.

Case 2. cr*D*(*G*∗) = 2 and choose the vertex notation of the graph *G*∗ presented as in Figure 1b. Since the set *RD* is nonempty and there is only one subdrawing of a subgraph *Fi* = *G*∗ ∪ *Ti* for all *Ti* ∈ *RD* represented by the rotation (13524), the subgraph *Ti* is crossed at least four times by edges of each subgraph *Tj* ∈ *RD* with *j* = *i*. Hence, by fixing the graph *G*∗ ∪ *Ti*

$$\operatorname{crr}\_D(G^\* + D\_n) \ge 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 4(r-1) + 2(n-r) + 2 = 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor$$

$$+ 2n + 2r - 2 \ge 4\left\lfloor \frac{n-1}{2} \right\rfloor \left\lfloor \frac{n-2}{2} \right\rfloor + 2n + 2\left(\left\lceil \frac{n}{2} \right\rceil + 1\right) - 2 \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

For all these mentioned cases, it turned out that there is no drawing of the graph *G*∗ + *Dn* with fewer than *n*2 − 2*n* + - *n* 2 . crossings, and the proof of Theorem 2 is complete.

**Figure 8.** Five possible components of the subgraph G*D* in which there are two adjacent edges with weights 1. Green, blue, brown, and black correspond to the values 1, 2, 3, and 4, respectively. (**a**): the complete graph *K*3 with edge weights 1, 1, and 2; (**b**): the complete graph *K*4 with edge weights 1, 1, 2, 2, 2, and 3; (**c**): the complete graph *K*4 with edge weights 1, 1, 1, 2, 2, and 3; (**d**): the complete graph *K*5; (**e**): the complete graph *K*6.

#### **5. Conclusions**

Into both drawings in Figures 4 and 5, we could add *n* − 1 or *n* edges forming paths *Pn*, *n* ≥ 2 or cycles *Cn*, *n* ≥ 3 on vertices of *Dn* with no crossing, respectively. Thus, the following surprising results are obvious.

**Corollary 3.** cr(*G*∗ + *Pn*) = *n*2 − 2*n* + - *n* 2 . *for n* ≥ 2*, i.e.,* cr(*G*∗ + *Pn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . *for n even and* cr(*G*∗ + *Pn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . − 1 *for n odd.*

**Corollary 4.** cr(*G*∗ + *Cn*) = *n*2 − 2*n* + - *n* 2 . *for n* ≥ 3*, i.e.,* cr(*G*∗ + *Cn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . *for n even and* cr(*G*∗ + *Cn*) = 4 - *n* 2 .- *n*−1 2 . + - *n* 2 . − 1 *for n odd.*

These results extend already known results of join products of graphs on at most six vertices with paths and cycles, see [2,5,18,20,26,33–41].

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


**1.1 Nitrogen uptake and assimilation**

Among the mineral nutrient elements, nitrogen is a kind of macronutrient. Most plant species are able to absorb and assimilate nitrate (NO3–), ammonium (NH4+), urea and amino acids as nitrogen sources. Most soils do not have sufficient N in available form to support desired production levels. Therefore, addition of N from fertilizer is typically needed to maximize crop yields. Many kinds of N fertilizers are used which contain varying forms of N such as NO3 ––N, NH4+–N and urea. However, NO3– form of nitrogen is the predominant form of N absorbed by plants, regardless of the source of applied N (Breteler and Luczak, 1982). This preference is due to several autotrophic soil bacteria, which rapidly oxidize NH4+ to NO2 –, and then to NO3– in warm, well–aerated soils. Even though NO3– is the most available form of N to plants, it can be more readily lost from the root zone because it is very mobile and easy to leach. This economically and environmentally undesirable process perpetuates a large amount of the uncertainty associated with N fertilizer management (Pessarakli, 2002).

In the soil solution, nitrate is carried towards the root by bulk flow and is absorbed into the epidermal and cortical symplasm. Within the root symplasm, nitrate has four fates: (1) reduced to nitrite by the cytoplasmic enzyme nitrate reductase; (2) efflux back across the plasma membrane to the apoplasm; (3) influx and stored in the vacuole; or (4) transported to the xylem for long−distance translocation to the leaves (Andrews, 1986; Ashley et al., 1975; Black et al., 2002; Cooper and Charkson, 1989). Translocated from the xylem, nitrate enters the leaf apoplasm to reach leaf mesophyll cells, where nitrate is again absorbed and either reduced to nitrite or stored in the vacuole.

Nitrate translocated from the roots through the xylem is absorbed by a mesophyll cell via one of the nitrate−proton symporters into the cytoplasm, reduced to nitrite by nitrate reductase (NR) in the cytoplasm, and then reduced to ammonium by nitrite reductase (NiR)

\* Corresponding author

Effect of Mixed Amino Acids on Crop Growth 121

consume plant material with rich nitrate, they may suffer from methemoglobinemia or carcinoma by converting nitrate to nitrite of nitrosamines. Some countries limit the nitrate

Several authors reported that free amino acids could down regulate nitrate uptake and nitrate content in plant. It was found that exogenously supplied amino acids and amides could decrease the uptake of nitrate by soybean (Muller and Touraine, 1992); wheat (Rodgers and Barneix, 1993); maize (Ivashikina and Sokolov, 1997; Padgett and Leonard, 1993, 1996; Sivasankar et al., 1997); barley (Aslam et al., 2001)(Table 1). Plants appear to have multiple mechanisms for regulating nitrate uptake in addition to amino acids or N-

Work by Breteler and Arnozis (1985) determined that pretreatment of dwarf bean roots with many different individual amino acids inhibited nitrate uptake to varying degrees dependent upon prior exposure of the plants to nitrogen and the specific amino acid treatment. No significant effect of amino acids on nitrate transport was detected when both NO3− and amino acids were present in the bathing solution, and no correlation emerged between inhibition of nitrate uptake and inhibition of nitrate reductase relative to specific amino acids. A more detailed study, presented by Muller and Touraine (1992), demonstrated inhibition of uptake by 50% or greater by alanine, glutamine, asparagines, arginine, β–alanine and serine when soybean seedlings were pretreated for 18 h prior to exposure to NO3−. The mechanisms of inhibition by arginine and alanine appeared to differ, however. Arginine stopped NO3− uptake immediately upon introduction to the uptake solution, kinetically similar to NH4+ inhibition. The authors suggested that this may be the result of a non-metabolic response such as alteration of membrane potentials. Inhibition by alanine was slower to develop, suggesting a metabolic component to the regulation rather

**treatment NO3− supplied NO3− uptake Remarks**

100 0.5 40−120 (Muller, 1992)

1.0 0.3 89−106 Non−starvation

starvation

(Rodger, 1993)

1.0 0.3 50−105 3 days N

**(mM) (%)**  Soybean 10 0.5 5−85 14 amino acids

Maize root 15 5.0 84 (Padgett, 1993) Barley root 1.0 0.1 40−50 (Aslam, 2001) 1.0 10 70

The N status of the plants could also affect the inhibitory effect of amino acids on nitrate uptake. Rodger and Barneix (1993) had supplied amino acids exogenously to N starved or non−starved wheat seedlings. Exogenously supplied amino acids and amides had no effect on the wheat seedlings under well nourishment. However, some of the amino acids and amides supplied seedlings starved of N for 3 days inhibited up to 50% of the nitrate uptake rate.

content in plant material sold for human consumption.

status (Padgett and Leonard, 1993).

than a physical or chemical interference.

**Amino acid**

Table 1. Effect of amino acid on NO3− uptake in several plants

**Plant materials**

Wheat

in the chloroplast, which is then incorporated into amino acids by the glutamine synthetase−glutamine− 2−oxoglutarate amidotransferase (GS/GOGAT) enzyme system, giving rise to glutamine (Gln) and ultimately other amino acids and their metabolites (Fig. 1; Taiz and Zeiger, 2002). Therefore, NR, NiR and GS constitute the first three enzymes of the nitrate assimilatory pathway. The NR activity is the limiting step of NO3 −–N conversion to amino acid synthesis (Campbell, 1999). In most plant species only a proportion of the absorbed nitrate is assimilated in the root, the remainder being transported upwards through the xylem for assimilation in the shoot where it is reduced and incorporated into amino acids (Forde, 2000).

Fig. 1. The main process of nitrate assimilation.