A Combined Logarithmic Bound on the Chromatic Index of a Multigraph

TL;DR

This paper establishes a logarithmic upper bound on the chromatic index of multigraphs, improving understanding of how much it can exceed the fractional chromatic index.

Contribution

It proves a new logarithmic bound on the chromatic index relative to the fractional index for multigraphs, refining previous bounds by Kahn.

Findings

Chromatic index is less than phi(G) + log(min{(n+1)/3, phi(G)})

The bound applies to multigraphs with at least one edge and order n > 3

Provides a tighter understanding of the relationship between chromatic and fractional chromatic indices.

Abstract

For a multigraph G, the integer round-up phi(G) of the fractional chromatic index yields a good general lower bound for the chromatic index . For an upper bound, Kahn showed that for any real c > 0 there exists a positive integer N so that the chromatic index is less than (1+c)*phi(G) whenever the fractional index > N. We show the amount by which the chromatic index can surpass phi(G) is in fact logarithmic, by showing that for any multigraph G with order n > 3 and at least one edge, the chromatic index is less than phi(G) + log (min {(n+1)/3, phi(G)}) .