### On Ramsey numbers for trees versus fans of even order

#### Abstract

Given two graphs *G* and *H*. The graph Ramsey number *R*(*G*, *H*) is the least natural number *r* such that for every graph *F* on *r* vertices, either *F* contains a copy of *G* or F̅ contains a copy of *H*. A vertex *v* is called a dominating vertex in a graph *G* if it is adjacent to all other vertices of *G*. A wheel *W*_{m} is a graph consisting one dominating vertex and *m* other vertices forming a cycle. A fan graph *F*_{1,m} is a graph formed from a wheel *W*_{m} by removing one cycle-edge. In this paper, we consider the graph Ramsey number *R*(*T*_{n},*F*_{1,m}) of a tree *T*_{n} versus a fan *F*_{1,m}. The study of *R*(*T*_{n},*F*_{1,m}) has been initiated by Li et. al. (2016) where *T*_{n} is a star, and continued by Sherlin et. al. (2023) for *T*_{n} which is not a star and fan *F*_{1,m} with even *m* ≤ 8. This paper will give the graph Ramsey numbers *R*(*T*_{n},*F*_{1,m}) for odd *m* ≤ 8.

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#### Full Text:

PDFDOI: http://dx.doi.org/10.19184/ijc.2024.8.1.2

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