The Ramsey Numbers of Paths Versus Wheels: a Complete Solution

Binlong Li,Bo Ning

doi:10.37236/3968

Abstract

Let $G_1$ and $G_2$ be two given graphs. 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 $\overline{G}$ contains a $G_2$. We denote by $P_n$ the path on $n$ vertices and $W_m$ the wheel on $m+1$ vertices. Chen et al. and Zhang determined the values of $R(P_n,W_m)$ when $m\leq n+1$ and when $n+2\leq m\leq 2n$, respectively. In this paper we determine all the values of $R(P_n,W_m)$ for the left case $m\geq 2n+1$. Together with Chen et al.'s and Zhang's results, we give a complete solution to the problem of determining the Ramsey numbers of paths versus wheels.

Highlights

We use Bondy and Murty [2] for terminology and notation not defined here, and consider finite simple graphs only.Let G be a graph
We denote by Pn the path on n vertices and Wm the wheel on m + 1 vertices
Together with Chen et al.’s and Zhang’s results, we give a complete solution to the problem of determining the Ramsey numbers of paths versus wheels

Summary

Introduction

We use Bondy and Murty [2] for terminology and notation not defined here, and consider finite simple graphs only. (this question of determining Ramsey numbers of paths versus paths appeared in a paper of Erdos [5] in 1947, and the right upper bound was determined there.) After that, Faudree et al [8] determined the Ramsey numbers of paths versus cycles. We list these results as bellow, both of them will be used in this paper. Graph theorists have begun to investigate the Ramsey numbers of paths versus wheels. Together with Theorems 3 and 4, we give a complete solution to the problem of determining the Ramsey numbers of paths versus wheels

Preliminaries

Proof of Theorem 5

Remarks