Ramsey Numbers of Odd Cycles Versus Larger Even Wheels

TL;DR

This paper proves a long-standing conjecture on the exact Ramsey numbers for odd cycles versus larger even wheels for most remaining cases, confirming that $R(C_{2k+1}, W_{2j})=4j+1$ under specific conditions.

Contribution

It establishes the exact value of the Ramsey number $R(C_{2k+1}, W_{2j})$ for nearly all remaining unresolved cases, advancing understanding of cycle and wheel Ramsey numbers.

Findings

Proves the conjecture for $j-k extgreater= 251$, $k<j<3k/2$, and $j extgreater= 212299$.

Confirms the exact Ramsey number $R(C_{2k+1}, W_{2j})=4j+1$ in these cases.

Extends previous asymptotic results to exact values for a broad class of parameters.

Abstract

The generalized Ramsey number $R(G_1, G_2)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1}, W_{n})$ of odd cycles versus larger wheels, leaving open the particular case where $n = 2j$ is even and $k<j<3k/2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1}, W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1}, W_{2j}) \le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j-k \ge 251$, $k<j<3k/2$, and $j \ge 212299$.