On Neutral Edge Sets in Anti-Ramsey Numbers

TL;DR

This paper investigates the conditions under which certain edges in a graph are neutral for the anti-Ramsey number, providing new insights into how these numbers behave for specific graph structures and proposing broader conjectures.

Contribution

It identifies specific parameter ranges where internal edges of certain paths are neutral for the anti-Ramsey number, extending known results and offering a complete determination for these cases.

Findings

Neutrality of internal edges for certain parameters

Complete determination of anti-Ramsey numbers for specific graphs

Conjecture on generalization of neutrality to broader cases

Abstract

The anti-Ramsey number of a graph $G$, introduced by Erd\H{o}s et al.\ in 1975, is the maximum number of colors in an edge-coloring of the complete graph $K_n$ that avoids a rainbow copy of $G$. We call a subset of edges of $G$ \emph{neutral} for the anti-Ramsey number if removing them does not alter the anti-Ramsey number of $G$. Let $k$, $t$, and $n$ be positive integers, and consider $G = kP_4 \cup tP_2$. Assume $S \subseteq E(G)$ consists of internal edges of the $P_4$ components in $G$. It is known that $S$ is neutral when $t \geq k+1 \geq 2$ and $n \geq 8k + 2t - 4$. In this paper, we identify values of $k \geq t$ such that, for all $n$ in a specific subinterval of $[8k + 2t - 4, \infty)$, $S$ remains neutral. Since the anti-Ramsey numbers for matchings are well understood, our results provide a complete determination of the anti-Ramsey number for $G$ under these conditions. Based on our findings, we conjecture that this neutrality may extend to the general case $t \geq 1$, $k \geq 1$, and $n \geq 4k + 2t$, but not when $t = 0$, $k \geq 2$, and $n \geq 4k$.