Degree bipartite Ramsey numbers

TL;DR

This paper investigates degree bipartite Ramsey numbers, establishing linear bounds for certain complete bipartite graphs and determining these numbers for trees and specific bipartite graphs.

Contribution

It introduces new bounds for degree bipartite Ramsey numbers and computes these values for trees, stars, paths, and complete bipartite graphs.

Findings

$r_{ ext{Δ}}(K_{m,n};s)$ is linear in $n$ for fixed $m$

Determined $br_{ ext{Δ}}(G;s)$ for trees, stars, and paths

Provided exact values for complete bipartite graphs

Abstract

Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\; \mbox{and} \;\chi(H)=2\}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with $m$ fixed. We also determine $br_{\Delta}(G;s)$ where $G$ are trees, including stars and paths, and complete bipartite graphs.