On the Anti-Ramsey Number Under Edge Deletion

According to a study by Erdős 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 $G=k{P}_4\cup t{P}_2$. Let $E^{\mathrm{\prime }}$ 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) $t\ge k+1\ge 2$ and $n\ge 8k+2t-4$; (ii) $k,t\ge 1$ and $n=4k+2t$; (iii) $k=1$, $t\ge 1$, and $n\ge 2t+4$, occurs then the behavior of the anti-Ramsey number remains consistent when the edges in $E^{\mathrm{\prime }}$ are removed from $G$, i.e., $AR(n,G)=AR(n,G-E^{\mathrm{\prime }})$. However, this is not the case when $k\ge 2$, $t=0$, and $n=4k$. As a result, we calculate $AR(k{P}_4\cup t{P}_2)$ for the cases: (i) $t\ge k+1\ge 2$ and $n\ge 8k+2t-4$; (ii) $k,t\ge 1$ and $n=4k+2t$; (iii) $k=1$, $t\ge 0$, and $n\ge 2t+4$; (iv) $k\ge 1$, $t=0$, and $n=4k$.