Edge-colouring and total-colouring chordless graphs

TL;DR

This paper proves that chordless graphs with maximum degree at least 3 can be optimally edge-colored and total-colored in polynomial time, confirming their chromatic index and total chromatic number.

Contribution

It establishes that all chordless graphs with maximum degree ≥ 3 have chromatic index equal to their maximum degree and total chromatic number one more, with polynomial-time algorithms.

Findings

Chordless graphs with Δ ≥ 3 have chromatic index Δ.

Chordless graphs with Δ ≥ 3 have total chromatic number Δ + 1.

The coloring algorithms are polynomial-time.

Abstract

A graph $G$ is \emph{chordless} if no cycle in $G$ has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree $\Delta\geq 3$ has chromatic index $\Delta$ and total chromatic number $\Delta + 1$. The proofs are algorithmic in the sense that we actually output an optimal colouring of a graph instance in polynomial time.