Complexity of computing the anti-Ramsey numbers for paths

Abstract

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdős, Simonovits and Sós. For given graphs $G$ and $H$ the anti-Ramsey number $ar(G,H)$ is defined to be the maximum number $k$ such that there exists an assignment of $k$ colors to the edges of $G$ in which every copy of $H$ in $G$ has at least two edges with the same color.

Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when $H$ is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number $ar(G,P_k)$, where $P_k$ is a path of length $k$. First, we observe that when $k$ is close to $n$ (the number of vertices in $G$), the problem is hard; hence, the challenging part is the computational complexity of the problem when $k$ is a fixed constant.

We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing $ar(G,P_k)$ for every integer $k\ge 3$ is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant $k$.