Generalised Ramsey numbers for two sets of cycles

TL;DR

This paper determines several generalized Ramsey numbers for pairs of cycle sets, extending classical results and providing new characterizations of critical graphs, especially when small cycles are involved.

Contribution

It extends previous work by calculating new generalized Ramsey numbers for cycle sets and characterizes most critical graphs for cycles of length up to five.

Findings

Determined all generalized Ramsey numbers where sets contain small or even cycles.

Provided a conjecture for the general case of cycle sets.

Characterized most critical graphs avoiding certain cycle lengths.

Abstract

We determine several generalised Ramsey numbers for two sets $\Gamma_1$ and $\Gamma_2$ of cycles, in particular, all generalised Ramsey numbers $R(\Gamma_1,\Gamma_2)$ such that $\Gamma_1$ or $\Gamma_2$ contains a cycle of length at most $6$, or the shortest cycle in each set is even. This generalises previous results of Erd\H{o}s, Faudree, Rosta, Rousseau, and Schelp from the 1970s. Notably, including both $C_3$ and $C_4$ in one of the sets, makes very little difference from including only $C_4$. Furthermore, we give a conjecture for the general case. We also describe many $(\Gamma_1,\Gamma_2)$-avoiding graphs, including a complete characterisation of most $(\Gamma_1,\Gamma_2)$-critical graphs, i.e., $(\Gamma_1,\Gamma_2)$-avoiding graphs on $R(\Gamma_1,\Gamma_2)-1$ vertices, such that $\Gamma_1$ or $\Gamma_2$ contains a cycle of length at most $5$. For length $4$, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which $|\Gamma_1|=|\Gamma_2|=1$. For lengths $3$ and $5$, our results are new even in this special case. Keywords: generalised Ramsey number, critical graph, cycle, set of cycles