Ramsey numbers of trees and unicyclic graphs versus fans

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≥m2−m+1. Zhang, Broersma and Chen showed that R(Sn,Fm)≥2n for n≤m2−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 UCn, which are connected graphs with n vertices and a single cycle. We prove that R(UCn,Fm)=2n−1 for all n≥m2−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.