On star-wheel Ramsey numbers

TL;DR

This paper determines the exact Ramsey number for the pair of a star graph and a wheel graph within a specific range of parameters, advancing understanding of graph Ramsey theory.

Contribution

The paper explicitly calculates the Ramsey number R(K_{1,n},W_m) for even m with n+2 ≤ m ≤ 2n-2, filling a gap in known results.

Findings

Exact value of R(K_{1,n},W_m) for specified parameters

Extension of known Ramsey number results to wheel and star graphs

Provides a precise formula for these Ramsey numbers

Abstract

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\bar{G}$ contains a $G_2$. In this note, we determined the Ramsey number $R(K_{1,n},W_m)$ for even $m$ with $n+2\leq m\leq 2n-2$, where $W_m$ is the wheel on $m+1$ vertices, i.e., the graph obtained from a cycle $C_m$ by adding a vertex $v$ adjacent to all vertices of the $C_m$.