On the Anti-Ramsey Number of Spanning Linear Forests with Paths of Lengths 2 and 3
Ali Ghalavand, Xueliang Li
TL;DR
We determine the anti-Ramsey number for the linear forest formed by $k$ copies of $P_3$ and $t$ copies of $P_2$ in the critical regime $n = 3k + 2t$. The main result states $AR(n, kP_3 \\cup tP_2) = \\frac{1}{2}(3k+2t-3)(3k+2t-4) + 1$ for $k \ge 1$, $t \ge 2$, proved by a two-case analysis and induction on $k$, aided by a lemma of He and Jin. The proof combines structural rainbow-subgraph arguments with counting bounds to show tightness and to rule out colorings exceeding the bound. This extends prior work that required $t$ to be quadratically large in $k$ and clarifies the anti-Ramsey behavior at the critical host size; it also leaves open the problem of $AR(n, kP_3 \\cup tP_2)$ for larger $n$ when $t$ is small.
Abstract
An edge-coloring of a graph $G$ assigns a color to each edge in the edge set $E(G)$. A graph $G$ is considered to be rainbow under an edge-coloring if all of its edges have different colors. For a positive integer $n$, the anti-Ramsey number of a graph $G$, denoted as $AR(n, G)$, represents the maximum number of colors that can be used in an edge-coloring of the complete graph $K_n$ without containing a rainbow copy of $G$. This concept was introduced by Erdős et al. in 1975. The anti-Ramsey number for the linear forest $kP_3 \cup tP_2$ has been extensively studied for two positive integers $k$ and $t$. Formulations exist for specific values of $t$ and $k$, particularly when $k \geq 2$, $t \geq \frac{k^2 - k + 4}{2}$, and $n \geq 3k + 2t + 1$. In this work, we present the anti-Ramsey number of the linear forest $kP_3 \cup tP_2$ for the case where $k \geq 1$, $t \geq 2$, and $n = 3k + 2t$. Notably, our proof for this case does not require any specific relationship between $k$ and $t$.