A Coloring Algorithm for $4K_1$-free line graphs

TL;DR

This paper presents a polynomial-time coloring algorithm for a subclass of graphs that excludes certain induced subgraphs, including $4K_1$-free line graphs, and relates to computing the chromatic index of graphs without large matchings.

Contribution

It introduces a polynomial-time coloring algorithm for a subclass of $Free$(claw, $4K_1$) graphs, including $4K_1$-free line graphs, advancing understanding of coloring complexities.

Findings

Polynomial-time coloring algorithm for $4K_1$-free line graphs.

Chromatic index computation for graphs without matchings of size four.

Extension of coloring results to broader graph classes.

Abstract

Let $L$ be a set of graphs. $Free$($L$) is the set of graphs that do not contain any graph in $L$ as an induced subgraph. It is known that if $L$ is a set of four-vertex graphs, then the complexity of the coloring problem for $Free$($L$) is known with three exceptions: $L $= {claw, $4K_1$}, $L$ = {claw, $4K_1$, co-diamond}, and $L$ = {$C_4$, $4K_1$}. In this paper, we study the coloring problem for $Free$(claw, $4K_1$). We solve the coloring problem for a subclass of $Free$(claw, $4K_1$) which contains the class of $4K_1$-free line graphs. Our result implies the chromatic index of a graph with no matching of size four can be computed in polynomial time.