Short fans and the 5/6 bound 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.