Injective edge-coloring of graphs with given maximum degree

TL;DR

This paper investigates the injective edge-coloring of graphs, establishing bounds on the injective chromatic index based on maximum degree and graph restrictions like girth and chromatic number, and compares these bounds with strong chromatic index bounds.

Contribution

It introduces bounds on the injective chromatic index considering girth and chromatic number constraints, extending understanding of injective edge-coloring in graphs.

Findings

Bounds on injective chromatic index in terms of maximum degree

Comparison with bounds on strong chromatic index

Analysis under girth and chromatic number restrictions

Abstract

A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $\chi'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index.