On the Anti-Ramsey Number Under Edge Deletion

TL;DR

This paper explores how the anti-Ramsey number of certain graphs behaves under edge deletion, identifying conditions where it remains unchanged and calculating exact values for various cases.

Contribution

It provides new insights into the stability of the anti-Ramsey number under edge deletion for specific graph classes, with exact calculations for multiple parameter scenarios.

Findings

Anti-Ramsey number remains stable under edge deletion for certain graph parameters.

Explicit formulas for AR(n, G) are derived for various cases.

Edge deletion can either preserve or alter the anti-Ramsey number depending on graph structure.

Abstract

According to a study by Erd\H{o}s et al. in 1975, the anti-Ramsey number of a graph \(G\), denoted as \(AR(n, G)\), is defined as the maximum number of colors that can be used in an edge-coloring of the complete graph \(K_n\) without creating a rainbow copy of \(G\). In this paper, we investigate the anti-Ramsey number under edge deletion and demonstrate that both decreasing and unchanging are possible outcomes. For three non-negative integers \(k\), \(t\), and \(n\), let \(G = kP_4 \cup tP_2\). Let \(E'\) be a subset of the edge set \(E(G)\) such that every endpoint of these edges has a degree of two in \(G\). We prove that if one of the conditions (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 1\), and \(n \geq 2t + 4\), occurs then the behavior of the anti-Ramsey number remains consistent when the edges in \(E'\) are removed from \(G\), i.e., \(AR(n, G) = AR(n, G - E')\). However, this is not the case when \(k \geq 2\), \(t = 0\), and \(n=4k\). As a result, we calculate \(AR(kP_4 \cup tP_2)\) for the cases: (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 0\), and \(n \geq 2t + 4\); (iv) \(k \geq 1\), \(t = 0\), and \(n = 4k\).