Constrained Ramsey numbers for rainbow $P_5$

TL;DR

This paper investigates the constrained Ramsey number for rainbow paths of length 4, confirming conjectures for various graph classes, and introduces new results and questions for further exploration in graph Ramsey theory.

Contribution

It extends known results by proving the conjecture for multiple classes of disconnected graphs with chromatic number at least 3, and explores bipartite variations and future research directions.

Findings

Confirmed the conjecture for disconnected graphs with chromatic number ≥ 3

Established new results for bipartite graph variations

Proposed multiple open questions for future research

Abstract

Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs $H$ and $G$, the {\it constrained Ramsey number} (also called {\it rainbow Ramsey number}) $f(H,G)$ is defined as the minimum integer $n$ such that, in every edge-coloring of $K_{n}$ with any number of colors, there is either a monochromatic copy of $H$ or a rainbow copy of $G$. Let $P_t$ be the path on $t$ vertices. Gy\'{a}rf\'{a}s, Lehel and Schelp proved that $f(H,P_5)=R_3(H)$ when $H$ is a path or a cycle. Li, Besse, Magnant, Wang and Watts conjectured that $f(H,P_5)=R_3(H)$ for any graph $H$, and confirmed this for all connected graphs and all bipartite graphs. In this paper, we address this conjecture for multiple classes of disconnected graphs with chromatic number at least 3. Our newly established general results encompass all known results on this problem. We also obtain several results for a bipartite variation of the problem. In addition, we propose a series of questions concerning this problem from multiple distinct aspects for further research.