Ramsey numbers for complete graphs versus generalized fans

TL;DR

This paper determines new exact values and bounds for Ramsey numbers involving complete graphs and generalized fan graphs, extending previous results to broader cases with large parameters.

Contribution

The paper generalizes known Ramsey number results to complete graphs versus multiple generalized fans, providing explicit formulas and bounds for large parameters.

Findings

Exact Ramsey number for K_m versus sF_{t,n} for large n

Exact Ramsey number for K_3 versus F_{3,n} and F_{4,n}

Upper bounds on Ramsey numbers involving generalized fans

Abstract

For two graphs $G$ and $H$, let $r(G,H)$ and $r_*(G,H)$ denote the Ramsey number and star-critical Ramsey number of $G$ versus $H$, respectively. In 1996, Li and Rousseau proved that $r(K_{m},F_{t,n})=tn(m-1)+1$ for $m\geq 3$ and sufficiently large $n$, where $F_{t,n}=K_{1}+nK_{t}$. Recently, Hao and Lin proved that $r(K_{3},F_{3,n})=6n+1$ for $n\geq 3$ and $r_{\ast}(K_{3},F_{3,n})=3n+3$ for $n\geq 4$. In this paper, we show that $r(K_{m}, sF_{t,n})=tn(m+s-2)+s$ for sufficiently large $n$ and, in particular, $r(K_{3}, sF_{t,n})=tn(s+1)+s$ for $t\in\{3,4\},n\geq t$ and $s\geq1$. We also show that $r_{\ast}(K_{3}, F_{4,n})=4n+4$ for $n\geq 4$ and establish an upper bound on $r(F_{2,m},F_{t,n})$.