Semistrong edge coloring and $(0,1)$-relaxed strong edge coloring of graphs

Abstract

In this work, we study two relaxations of the well-known strong edge coloring. A semistrong edge coloring of a graph G is an edge coloring in which every color class forms a matching M such that every edge of M is incident with at least one vertex of degree 1 in the subgraph of G induced by the vertices covered by M. For any two nonnegative integers s and t, an (s,t)-relaxed strong edge coloring of G is an edge coloring in which, for every edge e of G, at most s edges at distance 1 and at most t edges at distance 2 from e receive the same color as e. The corresponding chromatic indices are defined accordingly. We confirm a recent conjecture of Lužar, Mockovčiaková, and Soták [J. Graph Theory 105 (2024) 612-632], which asserts that every connected graph G with maximum degree Delta (>= 3), except for K_{Δ,Δ}, has a semistrong chromatic index at most Delta^2 - 1. This is achieved by constructing an edge coloring of G using at most Delta^2 - 1 colors that is simultaneously semistrong and (0,1)-relaxed strong. Consequently, every such graph also has (0,1)-relaxed strong chromatic index at most Delta^2 - 1.