Stability and Ramsey numbers for cycles and wheels

TL;DR

This paper investigates the structure of certain edge colorings in complete graphs avoiding specific cycles and wheels, providing structural results and asymptotic bounds for related Ramsey numbers.

Contribution

It introduces a structural decomposition for colorings avoiding cycles and wheels and establishes asymptotic bounds for Ramsey numbers involving these graphs.

Findings

Vertex-partition into three red and blue edge sets after removing at most two vertices

Bounds for Ramsey numbers of cycles versus wheels asymptotically matching conjectures

Structural characterization of colorings avoiding cycles and wheels

Abstract

We study the structure of red-blue edge colorings of complete graphs, with no copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n \ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any such coloring, deleting at most two vertices we obtain a vertex-partition of $G$ into three sets such that the edges inside the partition classes are red, and edges between partition classes are blue. As a second result, we obtain bounds for the Ramsey numbers of $r(C_{2k+1},W_{2j})$ for $k < j$ integers, which asymptotically confirm the values of $4j+1$, as it were conjectured by Zhang et al.