Injective Edge Chromatic Index of a Graph

TL;DR

This paper introduces the concept of the injective edge chromatic index, provides exact values and bounds for various graph classes, and proves the NP-completeness of determining this index.

Contribution

It defines the injective edge coloring number, computes it for several graph classes, and establishes its computational complexity as NP-complete.

Findings

Exact values of $ ext{chi}_i'(G)$ for specific graph classes

Upper and lower bounds for $ ext{chi}_i'(G)$

NP-completeness of deciding $ ext{chi}_i'(G)=k$

Abstract

Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq c(e_3)$. The injective edge coloring number $\chi_{i}^{'}(G)$ is the minimum number of colors permitted in such a coloring. In this paper, exact values of $\chi_{i}^{'}(G)$ for several classes of graphs are obtained, upper and lower bounds for $\chi_{i}^{'}(G)$ are introduced and it is proven that checking whether $\chi_{i}^{'}(G)= k$ is NP-complete.