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14
Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) ..."
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Cited by 1821 (20 self)
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least
Induced Subgraphs of Prescribed Size
"... A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let qG denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdo s and McKay we show that every graph G on n vertices for which qG#C log n contains an induced subgra ..."
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Cited by 6 (5 self)
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A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let qG denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdo s and McKay we show that every graph G on n vertices for which qG#C log n contains an induced subgraph with exactly y edges, for every y between 0 and n #C . Our methods enable us also to show that under much weaker assumption, i.e., qG#n=14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e .
CYCLE LENGTHS IN SPARSE GRAPHS
 COMBINATORICA
, 2008
"... Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of C(G)  over all graphs G of average degree d and girth g.Erdős [8] conjectured that C(G)=Ω ` d ⌊(g−1)/2⌋ ´ for all such graphs, and we prove this conjecture. In parti ..."
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Cited by 4 (0 self)
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Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of C(G)  over all graphs G of average degree d and girth g.Erdős [8] conjectured that C(G)=Ω ` d ⌊(g−1)/2⌋ ´ for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length Ω ` d ⌊(g−1)/2⌋ ´. The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle [7,11,21,24,25]. Moreover, our bound cannot be improved in general, since known constructions of dregular Moore Graphs of girth g have roughly that many vertices. We also show that Ω ` d ⌊(g−1)/2⌋ ´ is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in Hfree graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdős and Gyárfás [9] (see also Erdős [10]) that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an nvertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e O(log ∗ n) on the average degree of graph of order n with no cycle of length in the sequence, where log ∗ n is the number of times the binary logarithm must be applied to n to get a number which is at most one.
Cycle Spectra of Hamiltonian Graphs
, 2009
"... We prove that every Hamiltonian graph with n vertices and m edges has cycles of 4 at least ..."
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Cited by 1 (1 self)
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We prove that every Hamiltonian graph with n vertices and m edges has cycles of 4 at least
Dependent Random Choice
"... We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory ..."
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We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them.
LARGE NEARLY REGULAR INDUCED SUBGRAPHS
, 2008
"... For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular i ..."
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For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n,1) vertices. The problem of estimating f(n,1) was posed long ago by Erdős, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of f(n, c): (i) For fixed c>2.1, n 1−O(1/c) ≤ f(n, c) ≤ O(cn / log n). (ii) For fixed c =1+ε with ε>0 sufficiently small, f(n, c) ≥ n Ω(ε2 / ln(1/ε)). (iii) Ω(ln n) ≤ f(n, 1) ≤ O(n 1/2 ln 3/4 n). An analogous problem for not necessarily induced subgraphs is briefly considered as well.