Degree Ramsey numbers for even cycles

TL;DR

This paper investigates the degree Ramsey numbers for even cycles, establishing precise asymptotic bounds for specific cycles and improving lower bounds for general cases, advancing understanding in graph coloring and Ramsey theory.

Contribution

The paper provides exact asymptotic bounds for degree Ramsey numbers of certain even cycles and enhances lower bounds for general even cycles, filling gaps in existing knowledge.

Findings

R_Δ(C_6, s) = Θ(s^{3/2})

R_Δ(C_{10}, s) = Θ(s^{5/4})

Improved lower bounds for R_Δ(C_{2k}, s) for general k

Abstract

Let $H\xrightarrow{s} G$ denote that any $s$-coloring of $E(H)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_\Delta(G, s)$, is $\min \{\Delta(H): H \xrightarrow{s} G \}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_\Delta(C_{2k},s) \geq 2s$, and Kang and Perarnau showed that $R_\Delta(C_4, s) = \Theta(s^2)$. Our main result is that $R_\Delta(C_6, s) = \Theta(s^{3/2})$ and $R_\Delta(C_{10}, s) = \Theta(s^{5/4})$. Additionally, we substantially improve the lower bound for $R_\Delta(C_{2k}, s)$ for general $k$.