A local strengthening of Reed's {\omega}, \Delta, {\chi} conjecture for quasi-line graphs

TL;DR

This paper proves a local strengthening of Reed's conjecture for line and quasi-line graphs, providing efficient algorithms for coloring that meet these bounds, advancing understanding of graph coloring in these classes.

Contribution

It establishes the local strengthening of Reed's conjecture for line and quasi-line graphs and offers polynomial-time algorithms for optimal colorings.

Findings

Proves local strengthening for line graphs.

Extends results to all quasi-line graphs.

Provides faster coloring algorithms than previous methods.

Abstract

Reed's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi\leq \lceil\frac 12(\Delta+1+\omega)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a colouring that achieves the bound of Reed's original conjecture.