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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...
Expanding graphs and Ramsey numbers
, 1996
"... The generalized Ramsey number r(G; H) is investigated for H being a large order graph of bounded maximum degree. Giving a negative answer to a number of conjectures in generalized Ramsey theory it will be shown that for every nonbipartite graph G there is a function h(G; d) tending to infinity wi ..."
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The generalized Ramsey number r(G; H) is investigated for H being a large order graph of bounded maximum degree. Giving a negative answer to a number of conjectures in generalized Ramsey theory it will be shown that for every nonbipartite graph G there is a function h(G; d) tending to infinity with d, such that the Ramsey number r(G; H) ? h(G; d)jHj for almost every dregular graph H.