New bounds on the anti-Ramsey numbers of star graphs

Abstract

The anti-Ramsey number $ar(G,H)$ with input graph $G$ and pattern graph $H$, is the maximum positive integer $k$ such that there exists an edge coloring of $G$ using $k$ colors, in which there are no rainbow subgraphs isomorphic to $H$ in $G$. ($H$ is rainbow if all its edges get distinct colors). The concept of anti-Ramsey number was introduced by Erd\"os, Simanovitz, and S\'os in 1973. Thereafter several researchers investigated this concept in the combinatorial setting. Recently, Feng et al. revisited the anti-Ramsey problem for the pattern graph $K_{1,t}$ (for $t \geq 3$) purely from an algorithmic point of view due to its applications in interference modeling of wireless networks. They posed it as an optimization problem, the maximum edge $q$-coloring problem. For a graph $G$ and an integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of the graph $G$. Note that the optimum value of the edge $q$-coloring problem of $G$ equals $ar(G,K_{1,q+1})$. In this paper, we study $ar(G,K_{1,t})$, the anti-Ramsey number of stars, for each fixed integer $t\geq 3$, both from combinatorial and algorithmic point of view. The first of our main results presents an upper bound for $ar(G,K_{1,q+1})$, in terms of number of vertices and the minimum degree of $G$. The second one improves this result for the case of triangle-free input graphs. For a positive integer $t$, let $H_t$ denote a subgraph of $G$ with maximum number of possible edges and maximum degree $t$. Our third main result presents an upper bound for $ar(G,K_{1,q+1})$ in terms of $|E(H_{q-1})|$. All our results have algorithmic consequences.