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

Chandran, LS and Hashim, T and Jacob, D and Mathew, R and Rajendraprasad, D and Singh, N (2024) New bounds on the anti-Ramsey numbers of star graphs via maximum edge q-coloring. In: Discrete Mathematics, 347 (4).

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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�s 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. © 2024 Elsevier B.V.

Item Type:	Journal Article
Publication:	Discrete Mathematics
Publisher:	Elsevier B.V.
Additional Information:	The copyright for this article belongs toElsevier B.V.
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	04 Mar 2024 05:47
Last Modified:	04 Mar 2024 05:47
URI:	https://eprints.iisc.ac.in/id/eprint/84137

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