Asymptotics of the chromatic number for quasi-line graphs

It was proved by Kahn [J. Kahn, Asymptotics of the chromatic index for multigraphs, Journal of Combinatorial Theory Series A 68 (1996) 233–254] that the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph G we have χ ( G ) ≤ ( 1 + o ( 1 ) ) χ f ( G ) . We extend this result to quasi-line graphs; G is a quasi-line graph if the neighbourhood of any vertex of G can be covered by two cliques.