Paper

On the Anti-Ramsey Number Under Edge Deletion

Abstract

According to a study by Erdős et al. in 1975, the anti-Ramsey number of a graph

, 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

without creating a rainbow copy of

. 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

,

, and

, let

. Let

be a subset of the edge set \(E(G)\) such that every endpoint of these edges has a degree of two in

. We prove that if one of the conditions (i)

and

; (ii)

and

; (iii)

,

, and

, occurs then the behavior of the anti-Ramsey number remains consistent when the edges in

are removed from

, i.e., \(AR(n, G) = AR(n, G - E')\). However, this is not the case when

,

, and

. As a result, we calculate \(AR(kP_4 \cup tP_2)\) for the cases: (i)

and

; (ii)

and

; (iii)

,

, and

; (iv)

,

, and

.