Results 1  10
of
24
Explicit Ramsey graphs and orthonormal labelings
 THE ELECTRONIC J. COMBINATORICS
, 1994
"... We describe an explicit construction of trianglefree graphs with no independent sets of size m and with\Omega\Gamma m 3=2 ) vertices, improving a sequence of previous constructions by various authors. As a byproduct we show that the maximum possible value of the Lov'asz `function of a graph on ..."
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Cited by 38 (12 self)
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We describe an explicit construction of trianglefree graphs with no independent sets of size m and with\Omega\Gamma m 3=2 ) vertices, improving a sequence of previous constructions by various authors. As a byproduct we show that the maximum possible value of the Lov'asz `function of a graph on n vertices with no independent set of size 3 is \Theta(n 1=3 ), slightly improving a result of Kashin and Konyagin who showed that this maximum is at least\Omega\Gamma n 1=3 = log n) and at most O(n 1=3 ). Our results imply that the maximum possible Euclidean norm of a sum of n unit vectors in R n , so that among any three of them some two are orthogonal, is \Theta(n 2=3 ).
Bounding Ramsey numbers through large Deviation Inequalities
"... We develop a new approach for proving lower bounds for various Ramsey numbers, based on using large deviation inequalities. This approach enables us to obtain the bounds for the odiagonal Ramsey ); r < k, that match the best known bounds, obtained through the local lemma. We discuss also th ..."
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Cited by 22 (7 self)
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We develop a new approach for proving lower bounds for various Ramsey numbers, based on using large deviation inequalities. This approach enables us to obtain the bounds for the odiagonal Ramsey ); r < k, that match the best known bounds, obtained through the local lemma. We discuss also the bounds for a related Ramseytype problem and show, for example, that there exists a K free graph G on n vertices in which every cn log n vertices .
Tough Ramsey graphs without short cycles
 J. Algebraic Combinatorics
, 1995
"... A graph G is ttough if any induced subgraph of it with x> 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are ttough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeiche ..."
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Cited by 15 (2 self)
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A graph G is ttough if any induced subgraph of it with x> 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are ttough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a conjecture of Chvátal that there exists an absolute constant t0 so that every t0tough graph is pancyclic. The proof is by an explicit construction based on the tight relationship between the spectral properties of a regular graph and its expansion properties. A similar technique provides a simple construction of trianglefree graphs with independence number m on Ω(m 4/3) vertices, improving previously known explicit constructions by Erdős and by Chung, Cleve and Dagum.
On the Minimal Number of Edges in ColorCritical Graphs
, 2001
"... A graph G is kcritical if it has chromatic number k, but every proper subgraph of it is (k 1) colorable. This paper is devoted to investigating the following question: for given k and n, what is the minimal number of edges in a kcritical graph on n vertices, with possibly some additional restrict ..."
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Cited by 9 (3 self)
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A graph G is kcritical if it has chromatic number k, but every proper subgraph of it is (k 1) colorable. This paper is devoted to investigating the following question: for given k and n, what is the minimal number of edges in a kcritical graph on n vertices, with possibly some additional restrictions imposed? Our main result is that for every k 4 and n > k this number is at least k 1 2 + k 3 2(k 2 2k 1) n, thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges in kcritical graphs and provide some constructions of sparse kcritical graphs. A few applications of the results to Ramseytype problems and problems about random graphs are described. 1
Large K_rfree subgraphs in K_sfree graphs and some other Ramseytype problems
 RANDOM STRUCTURES & ALGORITHMS
, 2005
"... In this paper we present three Ramseytype results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers ..."
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Cited by 9 (7 self)
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In this paper we present three Ramseytype results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete graph Kr. Our first result provides a new lower bound for this problem, which improves previous results of various researchers. It also allows us to solve some special cases of a closely related question posed by Erdős. For two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any redblue coloring of the edges of the complete graph KN, contains either a red copy of G or a blue copy of H. The book with n pages is the graph Bn consisting of n triangles sharing one edge. Here we study the bookcomplete graph Ramsey numbers and show that R(Bn, Kn) ≤ O(n 3 / log 3/2 n), improving the bound of Li and Rousseau. Finally, motivated by a question of Erdős, Hajnal, Simonovits, Sós, and Szemerédi, we obtain for all 0 <δ<2/3 an estimate on the number of edges in a K4free graph of order n which has no independent set of size
Some Exact RamseyTurán Numbers
, 2011
"... Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi [8], the rRamseyTurán number of H, RTr(n,H,f(n)), is defined to be the maximum number of edges in an nvertex, Hfree graph G with αr(G) ≤ f(n) where αr(G) denotes the Krindependence number of G. ..."
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Cited by 2 (2 self)
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Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi [8], the rRamseyTurán number of H, RTr(n,H,f(n)), is defined to be the maximum number of edges in an nvertex, Hfree graph G with αr(G) ≤ f(n) where αr(G) denotes the Krindependence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RTr(n,Kr+s,o(n)) for every 2 ≤ s ≤ r. These constructions are sharp for an infinite family of pairs of r and s. The only previous sharp construction was by Bollobás and Erdős [6] for r = s = 2.
On small graphs with highly imperfect powers
 Discrete Mathematics
, 1992
"... Let an integer s ≥ 1 and a graph G be given. Let us denote by χs(G) the smallest integer χ for which there exists a vertexcolouring of G with χ colours such that any two distinct vertices of the same colour are at a distance greater than s. Let us denote by ωs(G) the maximal cardinality of a subs ..."
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Cited by 1 (1 self)
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Let an integer s ≥ 1 and a graph G be given. Let us denote by χs(G) the smallest integer χ for which there exists a vertexcolouring of G with χ colours such that any two distinct vertices of the same colour are at a distance greater than s. Let us denote by ωs(G) the maximal cardinality of a subset of the vertices of G with diameter at most s. Clearly χs(G) ≥ ωs(G). For s ≥ 1 and h ≥ 0 set γs(G) = χs(G) − ωs(G) and νs(h) = max {n ∈ N: for any graph G, G  < n implies γs(G) < h}. Gionfriddo [13] has given estimates for νs(h). We improve the recent bound ν2(h) ≤ 6h (h ≥ 3) of Gionfriddo and Milici [14] to ν2(h) ≤ 5h (h ≥ 3). More generally, we give the following tight bounds for arbitrary s ≥ 1 and large enough h: 2h + 1 3 √ 2 (h log h)1/2 ≤ νs(h) ≤ 2h + h 1−ɛs, where ɛs> 0 depends only on s. The upper bound is proved entirely by constructive methods.