New bounds on the anti-Ramsey numbers of star graphs via maximum edge q-coloring

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 Erdős et al. 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 K1,t (for t≥3) purely from an algorithmic point of view. For a graph G and an integer q≥2, an edge q-coloring of G is an assignment of colors to edges of G, such that the 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,K1,q+1). Here, we study ar(G,K1,t), the anti-Ramsey number of stars, for each fixed integer t≥3, both from combinatorial and algorithmic point of view. The first of our main results presents an upper bound for ar(G,K1,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. Our third main result presents an upper bound for ar(G,K1,q+1) in terms of |E(G≤(q−1))|, which is a frequently used lower bound for ar(G,K1,q+1) and maximum edge q-coloring in the literature. All our results have algorithmic consequences.