Abstract
Given two graphs $$G_1$$G1 and $$G_2$$G2, the Ramsey number $$R(G_1, G_2)$$R(G1,G2) is the smallest integer $$N$$N such that, for any graph $$G$$G of order $$N$$N, either $$G_1$$G1 is a subgraph of $$G$$G, or $$G_2$$G2 is a subgraph of the complement of $$G$$G. We consider the case that $$G_1$$G1 is a cycle and $$G_2$$G2 is a (generalized) wheel. We expand the knowledge on exact values of Ramsey numbers in three directions: large cycles versus wheels of odd order; large wheels versus cycles of even order; and large cycles versus generalized odd wheels.
Highlights
In this paper we deal with finite simple graphs only
For a nonempty proper subset S ⊆ V (G), we let G[S] and G − S denote the subgraphs induced by S and V (G)\S, respectively
For large wheels versus cycles of odd order, besides Theorem 2 on triangles versus wheels of arbitrarily large order, Zhou [31] showed that R(Wn, Cm) = 2n + 1 for m odd and n ≥ 5m − 7
Summary
In this paper we deal with finite simple graphs only. For any undefined terminology and notation we refer the reader to the textbook of Bondy and Murty [3]. For large wheels versus cycles of odd order, besides Theorem 2 on triangles versus wheels of arbitrarily large order, Zhou [31] showed that R(Wn, Cm) = 2n + 1 for m odd and n ≥ 5m − 7 Even though it has been cited many times, the correctness of the proof in this Chinese language paper is questionable. We can use known results on R(C4, K1,n) to obtain new values for R(C4, Wn) immediately For large cycles versus generalized odd wheels, in the final section we prove the following result that has the same flavor.
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