# On the anti-Ramsey number of forests

## TL;DR

This paper determines the exact and approximate values of the anti-Ramsey number for various forests in complete graphs, advancing understanding of rainbow subgraph avoidance in edge-colored graphs.

## Contribution

It provides exact values for star forests and double stars, and approximate values for linear forests, expanding the knowledge of anti-Ramsey numbers for forests.

## Key findings

- Exact anti-Ramsey number for star forests.
- Approximate anti-Ramsey number for linear forests.
- Exact value of ar(K_n, 2P_4) for n ≥ 8.

## Abstract

We call a subgraph of an edge-colored graph rainbow subgraph, if all of its edges have different colors. The anti-Ramsey number of a graph $G$ in a complete graph $K_{n}$, denoted by $ar(K_{n}, G)$, is the maximum number of colors in an edge-coloring of $K_{n}$ with no rainbow subgraph copy of $G$. In this paper, we determine the exact value of the anti-Ramsey number for star forests and the approximate value of the anti-Ramsey number for linear forests. Furthermore, we compute the exact value of $ar(K_{n}, 2P_{4})$ for $n\ge 8$ and $ar(K_{n}, S_{p,q})$ for large $n$, where $S_{p,q}$ is the double star with $p+q$ leaves.