Revisiting Semistrong Edge-Coloring of Graphs

TL;DR

This paper investigates semistrong edge-colorings in graphs, establishing tight bounds for general graphs and those with maximum degree 3, and connects these results to other graph problems.

Contribution

It provides new tight upper bounds for semistrong edge-colorings in general graphs and degree-3 graphs, advancing understanding of this coloring concept.

Findings

Established tight upper bounds for general graphs.

Derived bounds for graphs with maximum degree 3.

Connected semistrong edge-coloring bounds to other graph problems.

Abstract

A matching $M$ in a graph $G$ is {\em semistrong} if every edge of $M$ has an endvertex of degree one in the subgraph induced by the vertices of $M$. A {\em semistrong edge-coloring} of a graph $G$ is a proper edge-coloring in which every color class induces a semistrong matching. In this paper, we continue investigation of properties of semistrong edge-colorings initiated by Gy\'{a}rf\'{a}s and Hubenko ({Semistrong edge coloring of graphs}. \newblock {\em J. Graph Theory}, 49 (2005), 39--47). We establish tight upper bounds for general graphs and for graphs with maximum degree $3$. We also present bounds about semistrong edge-coloring which follow from results regarding other, at first sight non-related, problems. We conclude the paper with several open problems.