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