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A Conjecture of Erdös the Ramsey Number r(W6)
 J. Combinatorial Math. and Combinatorial Computing
, 1996
"... It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by ve ..."
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It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...
All Ramsey numbers r(K 3 ,G) for connected graphs of order 9
, 1998
"... We determine the Ramsey numbers r(K 3 , G) for all 261080 connected graphs of order 9 and further Ramsey numbers of this type for some graphs of order up to 12. Almost all of them were determined by computer programs which are based on a program for generating maximal trianglefree graphs. ..."
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We determine the Ramsey numbers r(K 3 , G) for all 261080 connected graphs of order 9 and further Ramsey numbers of this type for some graphs of order up to 12. Almost all of them were determined by computer programs which are based on a program for generating maximal trianglefree graphs.
Guangxi Academy of Science
"... Based on a study of basic properties of cyclic graphs of prime order, we give an algorithm for computing lower bounds of classical Ramsey numbers. Our algorithm reduces certain amount of computation of cyclic graphs of prime order, since only some of them normalized cyclic graphs require computatio ..."
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Based on a study of basic properties of cyclic graphs of prime order, we give an algorithm for computing lower bounds of classical Ramsey numbers. Our algorithm reduces certain amount of computation of cyclic graphs of prime order, since only some of them normalized cyclic graphs require computation in our method. Using the algorithm, we construct ten cyclic graphs of prime order to obtain new lower bounds of ten classical Ramsey numbers:
Ramsey Numbers of K m versus (n,k)graphs and the Local Density of Graphs not Containing a K m
"... In this paper generalized Ramsey numbers of complete graphs Km versus the set h; n; ki of (n; k)graphs are investigated. The value of r(Km ; hn; ki) is given in general for (relative to n) values of k small compared to n using a correlation with Tur'an numbers. These generalized Ramsey number ..."
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In this paper generalized Ramsey numbers of complete graphs Km versus the set h; n; ki of (n; k)graphs are investigated. The value of r(Km ; hn; ki) is given in general for (relative to n) values of k small compared to n using a correlation with Tur'an numbers. These generalized Ramsey numbers can be used to determine the local densities of graphs not containing a subgraph Km . 1 Introduction Let m, l, n and k be positive integers with 0 l i m 2 j and 0 k i n 2 j and let hn; ki denote the set of all (n; k)graphs, i.e. the set of all graphs with n vertices and k edges. The Ramsey number r(hm; li; hn; ki) is defined as the smallest integer p such that in every redgreen coloring of the edges of the complete graph K p a green (m; l)graph or a red (n; k)graph occurs, i.e. a green graph with m vertices and l edges or a red graph with n vertices and k edges. Note that r(hm; li; hn; ki) is the classical Ramsey number r(Km ; K n ) if l = i m 2 j and k = i n 2 j ...