On the Anti-Ramsey Number Under Edge Deletion

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 $n = 4k$.