Injective edge-coloring of graphs with small maximum degree

TL;DR

This paper establishes an upper bound of 7 for the injective chromatic index of graphs with maximum degree 4 and certain density constraints, advancing understanding of edge-coloring in such graphs.

Contribution

The paper proves that graphs with maximum degree 4 and mad less than 8/3 have an injective chromatic index at most 7, providing a new bound in graph coloring theory.

Findings

Proves $oldsymbol{ ext{chi}_i'}(G) oldsymbol{ ext{leq}} 7$ for graphs with $oldsymbol{ ext{max degree} oldsymbol{ ext{leq}} 4$ and mad less than 8/3.

Establishes a bound connecting maximum degree, mad, and injective edge-coloring.

Contributes to the theory of graph coloring by bounding the injective chromatic index under specific conditions.

Abstract

An injective $k$-edge-coloring of a graph $G$ is a mapping $\phi$: $E(G)\rightarrow\{1,2,...,k\}$, such that $\phi(e)\ne\phi(e')$ if edges $e$ and $e'$ are at distance two, or are in a triangle. The smallest integer $k$ such that $G$ has an injective $k$-edge-coloring is called the injective chromatic index of $G$, denoted by $\chi_i'(G)$. In this paper, we prove that $\chi_i'(G)\le 7$ for every graph $G$ with $\Delta(G)\leq 4$ and mad$(G)<\frac{8}{3}$, where $\Delta(G)$ is the maximum degree of $G$.