. G, is such that 58 B. This impl'es that a. Let us consider two cases according to the size of p. This probability space was defined in [ BoIlobLs, A.
For example, having selected a blue G, or G,,, we could have defined a sequence of coloured graphs H o, HI, Suppose we have defined Hi. Another possibility is to add s independent green edges to G,, each being incident with precisely one vertcx of dcgree less than k. In yet another variant, for each x i of degree less than k we can choose k--d x, random edges incident with xi. All one wants from these essentially equivalent models is that Lernina 2 should be true and the space 9 n, p ; 3 k should not be far from g n, p, in particular, that we should have a firm grip on the probabilities in the new space, in the vein of our next lemma.
G, E 9 n,p ; 2 k has the following properties. The asscrtions i n i follow from the fact that a x.
The expected numbcr of vertices of degree k - l! The expected number of vertices of G,, having degree k-l! G, every green edge is incident with a vertex of degree at least 4 log IZ. G, provided a. Clearly this is the case if Q is the property of being k-connected. The next result shows that a. By Lemma 2 all wehaveto prove is that a. Let us estimate the probability that K G, -log n For -log n. This can be proved by imitating the proof or Theorem 1 for the appropriate model.
This inloriuation I S g,ven in the remaining columns. Matchings Complete matchings in random graphs were studied in great detail by Erdos and Renyi [31],, Our first aim is to pr-ove an analogue of Theorem 4 for matchings in bipartite graphs. To formulate the result prccraely we need soiiie delinitions.
Finally, write nzutciz for the property of containing a complete matching. I I Theorem 5. Uollobas [7, pp.
click Tfornason 64 I Lemma 6. Suppose G has no isolated vertex and it does not have a complete matching. As G does not have a complete matcliing, Hall's condition is violated by some set A c V i, i. Choose a set A of smallest cardinality satisfying Then i and ii hold since otherwise A could be replaced by a proper subset of itself. What is the probability that the subgraph of G spanned by A, v A, has at least 2a-2 edges and no vertex of A l is joined to a vertex in V, - A,? G, has a matching covering all but at most one of the vertices.
Thus if n is even then a. G, has a complete matching. Our aim is to show that this is true for much smaller probabilities. Supposc a x. G, has a matching covering all but at most one of the vertices of degree at least 1. At least how large is p? Y, uncovered. It is easily seen that if alniost G, contains such a configuration of six vertices then. We shall need two simple lemmas. Lemma 7. Thomason Proof. Lemma 8. Theorem 9. Let A be the event that G fails to have a matching covering all but at most one of its vertices of non-zero degree. Thomason 68 The proof of Theorem 9 was somewhat cumbersome because we had to consider graphs with only about 4 4 log n edges.
As we know, about twice as many edges are needed to make it likely that a random graph has no isolated vertices. I n that range the proof would have been considerably simpler, as would a direct proof of the following immediate consequence of Theorem 9. C,hnsa I-jirctor. The obvious obstruction to a complete matching is the existence of three vertices, two of which have degree 1 and are joined to the third; for brevity in Tables 6 and 7 such a configuration is called a star.
Tables S and 9 are about graphs of order At each stage, if there were vertices of degree I, we took the edge incident with a vertex of degree 1 and deleted both vertices from thegraph.
Ifevery non-isolated vertex was incident with at least 2 edges, we picked one of the edges at random and deleted both vertices. Having come to an end, we recorded the number of edges missing from a perfect matching in the middle third of the tables and used alternating chains to extend it to a matching. The last third of the tables shows how long the chains were which we had to use to extend our inatching to a complete matching, providcd the graph did contain n complete matching.
Hamilton cycles Mucheffort has been put into the study of Hamilton cycles in random graphs.
Theorem Denote by Ham the property of containing a Hamilton cycle. When searching for a Hamilton cycle, the lollipop algorithm see Thomason [Sl] was found useless and we had to do an exhaustive search. It is somewhat surprising that the probability that the hitting times coincide does not seem to increase fast.
Keywords: graphs, hypergraphs, Ramsey theory, extremal graph theory,. Ramsey -Tur an.. or vector valued) random variables should be estimated. Find the.Today’s Sponsor: PayNearMe
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In This Show We Cover…
- Why Brie chose a triplex for her first primary residence
- Choosing appreciation vs. cash flow in a buy and hold investment
- Why Brie decided to diversify her investments into long-distance investing.
- What makes an area “landlord–friendly” vs. “tenant friendly“
- Dealing with maintenance request issues.. even while on vacation.
- Why Brie calls herself a “YouTube Landlord“
- Real Estate as a hobby vs. a business
- Why Brie encourages Pets.. even a 120lb dog!
- When should you quit your job?
- Investing with your spouse: how to make it work
- House Hacking and using your personal finances to fund your investments
- Using online software to help you control your spending and invest in real estate
- The one question that Brie asked that led to a ton of deals, financing, and partnerships.
- Should you allow smokers in your rental unit?
- And a TON more!
Links from the Show
- BiggerPockets Webinars
- PayNearMe.com/biggerpockets
- Handyman Matters
- Income Property TV Show
- How to Hack Your Housing and Get Paid to Live For Free Blog Post by Brandon Turner
- Mint.com
- The Book on Estimating Rehab Costs book by J Scott, Published by BiggerPockets Publishing.
- Tenant Screening: The Ultimate Guide
- Postlets.com
- IfThisThenThat: IfTTT.com
Books Mentioned in the Show:
- The Millionaire Real Estate Investor by Gary Keller
- The One Thing by Gary Keller
Tweetable Topics:
“They should force lawmakers to own rental properties before writing the laws.” (Tweet This!)
“BiggerPockets: Pissing Off Employers Since 2004” (Tweet This!)
“What’s more important? Getting rich or staying married?” (Tweet This!)
“The beginning of budgeting is understanding where you are.” (Tweet This!)
“I’ve learned more on the BiggerPockets forums than any book I’ve ever read.” (Tweet This!)
Connect with Brie:
Brie’s BiggerPocket Profile
Brie’s Twitter: @ChicagoBrie
Brie’s Website: ChicagoBrie.com
Thanks for cheering up my morning guys and gal!!
Huy- Thanks for listening!
Brie,
you mentioned in your podcast several excel spreadsheets you use for household budgeting. I use Mint, but cant seem to get any real traction on getting a real monthly spending budget. You mentioned you might share what you have?
Sure, I added it to the file place
Happy to help, Huy! Thanks for listening.
Make that a 2nd, @Huy Nguyen. Great Morning Show! Thanks.
Thanks Tom and Roni!
Woo hoo! Finally downloaded BP Podcasts onto my iPod! Now I can listen anywhere! Thanks, guys!
GK – Glad you got them downloaded and were able to listen!
Great show today guys. I really enjoyed hearing about Brie’s experiences and successes. You are really doing a great service to the REI community.
I was curious and maybe I missed it, do you have a threshold for how leveraged you get with your investments, 80% LTV, 50% LTV all cash?
By the way, I’m with Brandon, there is something unexplainable about a good Starbucks beverage.
Congrats and continued success.
I