On a ramsey-type problem of J. A. Bondy and P. Erdös. I
TLDR
In this paper, the smallest integer m for which the following statement is true was shown: if a G graph has at least m vertices, then either G contains a Cn (cycle of length n) or G contain a Cp (cyclic length n), then R(Cn, C2r) = n + r − 1.About:
This article is published in Journal of Combinatorial Theory, Series B.The article was published on 1973-08-01 and is currently open access. It has received 131 citations till now.read more
Citations
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Journal ArticleDOI
Small Ramsey Numbers
TL;DR: This work presents data which, to the best of its knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge.
Journal ArticleDOI
On the combinatorial problems which I would most like to see solved
TL;DR: I was asked to write a paper about the major unsolved problems in com-binatorial mathematics, but after some thought it seemed better to modify the title to a less pretentious one, so I state only my three favourite problems.
Journal ArticleDOI
The Ramsey number for a triple of long even cycles
Agnieszka Figaj,Tomasz Łuczak +1 more
TL;DR: It is shown that for any real positive numbers @a"1,@a"2, @ a"3 the Ramsey number for a triple of even cycles of lengths is (asymptotically) equal to (@a" 1+@a’s2+@A"3+max{-o(1),-1,-2,-3}+o( 1) .
Book ChapterDOI
Generalized ramsey theory for graphs - a survey
TL;DR: A survey of generalized Ramsey numbers for graphs can be found in this article, where the authors emphasize the following class of problems: given graphs G1,..., Gc, determine or estimate the Ramsey number r(G1,, Gc), the smallest number p such that if the lines of a complete graph Kp are c-colored in any manner, then for some j there exists a subgraph in color j which is isomorphic to Gj.
Journal ArticleDOI
The Ramsey number for hypergraph cycles I
Penny Haxell,Tomasz Łuczak,Yuejian Peng,Vojtěch Rödl,Andrzej Ruciński,Miklós Simonovits,Jozef Skokan +6 more
TL;DR: It is proved that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4.
References
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Ramsey Numbers for Cycles in Graphs
J.A Bondy,Paul Erdös +1 more
TL;DR: In this article, it was shown that the Ramsey number R(C n, C n ) = 2n-1 if n is odd, if n > r(2r-1) and if n ≥ 4r 2 -r+2.
Journal ArticleDOI
On the Existence of Specified Cycles in Complementary Graphs.
Gary Chartrand,Seymour Schuster +1 more
TL;DR: In this article, the least integer p such that for any graph G of order p, either G has an m-cycle or its complement G bar has an n-cycle was established for m,n = or < 6.