On a fractional version of Haemers' bound

TL;DR

This paper introduces a fractional version of Haemers' bound on the Shannon capacity of graphs, which outperforms traditional bounds and is multiplicative, offering a new tool for graph capacity analysis.

Contribution

It proposes a novel fractional Haemers' bound that strengthens existing bounds and demonstrates its multiplicative property, unlike the original Haemers' bound.

Findings

Fractional Haemers' bound outperforms classical bounds on Shannon capacity.

The new bound is multiplicative, unlike Haemers' original bound.

The bound combines strengths of Haemers' bound and fractional chromatic number.

Abstract

In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers' bound. We show also that this bound is multiplicative, unlike Haemers' bound.