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131.
132.
Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C k denote a cycle of length k, and let C k denote the set of cycles C ?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C k ) ~ z(n,F) — here we write f(n) ~ g(n) for functions f,g: ? → ? if lim n→∞ f(n)/g(n)=1. They proved this when F ={C 4} by showing that ex(n,{C 4;C 5})~z(n,C 4). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C 2? ∪{C k }) ~ z(n,C 2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C 2? ∪{C k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest. 相似文献
133.
In this short note, we prove that for β<1/5 every graph G with n vertices and n2−β edges contains a subgraph G′ with at least cn2−2β edges such that every pair of edges in G′ lie together on a cycle of length at most 8. Moreover edges in G′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdős, and Rödl. 相似文献
134.
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χg(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph Gn,p. We show that with high probability, the game chromatic number of Gn,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 相似文献
135.
136.
We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c ? o(1))n. Moreover, there is a subgraph G′′of G with (1/2 + γ ? o(1))|E| edges that does not contain a cycle of length at least cn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012 相似文献
137.
In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability ai,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected. 相似文献
138.
A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least \begin{align*}\left\lceil n/2 \right\rceil\end{align*} is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if p ? log n/n, then a.a.s. every subgraph of G(n,p) with minimum degree at least (1/2 + o (1) )np is Hamiltonian. Our result improves on previously known bounds, and answers an open problem of Sudakov and Vu. Both, the range of edge probability p and the value of the constant 1/2 are asymptotically best possible. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 相似文献
139.
The classical result of Erd?s and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \begin{align*}p=\frac{1}{n}\end{align*} – for any ε > 0 and \begin{align*}p=\frac{1-\epsilon}{n}\end{align*}, all connected components of G(n,p) are typically of size Oε(log n), while for \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 相似文献
140.
Jacob?FoxEmail author Choongbum?Lee Benny?Sudakov 《Israel Journal of Mathematics》2012,191(2):959-971
An old problem of Moser asks: what is the size of the largest union-free subfamily that one can guarantee in every family of m sets? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. We show that every family of m sets contains a union-free subfamily of size at least \(\left\lfloor {\sqrt {4m + 1} } \right\rfloor - 1\) and that this bound is tight. This solves Moser’s problem and proves a conjecture of Erd?s and Shelah from 1972.More generally, a family of sets is a-union-free if there are no a + 1 distinct sets in the family such that one of them is equal to the union of a others. We determine up to an absolute multiplicative constant factor the size of the largest guaranteed a-union-free subfamily of a family of m sets. Our result verifies in a strong form a conjecture of Barat, Füredi, Kantor, Kim and Patkos. 相似文献