Asymptotics of the chromatic number for quasi-line graphs

TL;DR

This paper extends the asymptotic equivalence of chromatic and fractional chromatic numbers from line graphs to quasi-line graphs and provides a polynomial-time algorithm to approximate the chromatic number.

Contribution

It generalizes Kahn's asymptotic result to quasi-line graphs and introduces a polynomial-time approximation algorithm for their chromatic number.

Findings

Asymptotic equality of chromatic and fractional chromatic numbers for quasi-line graphs

Polynomial-time algorithm for near-optimal coloring of quasi-line graphs

Extension of known results from line graphs to a broader class of claw-free graphs

Abstract

As proved by Kahn, the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph $G$ we have $\chi(G) \leq (1+o(1))\chi_f(G)$. We extend this result to quasi-line graphs, an important subclass of claw-free graphs. Furthermore we prove that we can construct a colouring that achieves this bound in polynomial time, giving us an asymptotic approximation algorithm for the chromatic number of quasi-line graphs.