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

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 Erdos 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 K-1,K-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, K-1,K-q+1). Here, we study ar(G, K-1,K-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, K-1,K-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, K-1,K-q+1) in terms of |E(G(<=(q-1)))|, which is a frequently used lower bound for ar(G, K-1,K-q+1) and maximum edge q -coloring in the literature. All our results have algorithmic consequences. (c) 2024 Elsevier B.V. All rights reserved.