Rainbow short linear forests in edge-colored complete graph

Abstract

An edge-colored graph $G$ is called rainbow if no two edges of $G$ have the same color. For a graph $G$ and a subgraph $H\subseteq G$, the anti-Ramsey number $AR(G,H)$ is the maximum number of colors in an edge-coloring of $G$ such that $G$ contains no rainbow copy of $H$. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this direction. We refine the bound and obtain the precise value of $AR(K_n,P_3\cup t{P}_2)$ for all $n\ge 2t+3$. Additionally, we determine the value of $AR(K_n,2P_3\cup t{P}_2)$ for any integers $t\ge 1$ and $n\ge 2t+7$.