Short fans and the 5/6 bound for line graphs

TL;DR

This paper proves a conjecture about the chromatic number of line graphs being bounded by a function of clique number and maximum degree, and introduces techniques relevant to broader graph coloring problems.

Contribution

It confirms the 2011 conjecture on line graph coloring bounds and develops new methods potentially useful for the Goldberg--Seymour conjecture.

Findings

Proved the conjecture that hi(G)  max{ (G), (5   (G)+8)/6} for line graphs.

Developed general techniques for graph coloring that may impact related conjectures.

Enhanced understanding of the chromatic bounds for line graphs.

Abstract

In 2011, the second author conjectured that every line graph $G$ satisfies $\chi(G)\le \max\{\omega(G),\frac{5\Delta(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture.