Ramsey numbers of trees and unicyclic graphs versus fans

Abstract

Abstract. The generalized Ramsey number R(H, K) is the smallest positive integer n such that for any graph G with n vertices either G contains H as a subgraph or its complement G contains K as a subgraph. Let Tn be a tree with n vertices and Fm be a fan with 2m + 1 vertices consisting of m triangles sharing a common vertex. We prove a conjecture of Zhang, Broersma and Chen for m ≥ 9 that R(Tn, Fm) = 2n − 1 for all n ≥ m 2 − m + 1. Zhang, Broersma and Chen showed that R(Sn, Fm) ≥ 2n for n ≤ m 2 − m where Sn is a star on n vertices, implying that the lower bound we show is in some sense tight. We also extend this result to unicyclic graphs U Cn, which are connected graphs with n vertices and a single cycle. We prove that R(U Cn, Fm) = 2n − 1 for all n ≥ m 2 −m+1 where m ≥ 18. In proving this conjecture and extension, we present several methods for embedding trees in graphs, which may be of independent interest.

Ramsey Numbers of Trees Versus Odd Cycles

“…The next claim is the key ingredient to show that there is another large set S 2 analogous to S 1 and disjoint from S 1 in G. The proof of this claim applies a method to greedily embed trees used in the proof of Claim 3.3 in [2].…”

“…The next claim is the key ingredient to show that there is another large set S 2 analogous to S 1 and disjoint from S 1 in G. The proof of this claim applies a method to greedily embed trees used in the proof of Claim 3.3 in [2]. We now define a procedure to greedily extend this embedding of H to an embedding of T n in G. At any point in this procedure, let K denote the subgraph of G induced by the set of vertices that have so far been embedded to.…”

“…The proof of the next lemma is from our work on Ramsey numbers of trees versus fans as in [2] and is included here for completeness. Lemma 3.…”

Ramsey Numbers of Large Even Cycles and Fans

“…For more Ramsey numbers involving fans, we refer the reader to [9,22,25,30], etc. In this paper, we are concerned with the asymptotic behavior of the Ramsey number R(C 2 an , F n ) when n is large and a 1/2 is fixed.…”