The Ramsey numbers of fans versus a complete graph of order five

Abstract

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that for any graph $G$ of order $N$, either $G$ contains $F$ or the complement of $G$ contains $H$. Let $F_l$ denote a fan of order $2l+1$, which is $l$ triangles sharing exactly one vertex, and $K_n$ a complete graph of order $n$. Surahmat et al. conjectured that $R(F_l,K_n)=2l(n-1)+1$ for $l\geq n\geq 5$. In this paper, we show that the conjecture is true for n=5.