The Fractional Haemers Bound of The Mycielski Construction

TL;DR

This paper studies how the generalized Mycielski construction affects the complementary fractional Haemers bound of a graph, extending known results for fractional chromatic number and Lovász theta number, and providing bounds and conditions for equality.

Contribution

It extends Tardif's formula for fractional chromatic number to the fractional Haemers bound under certain conditions and provides tight bounds for the effect of the Mycielski construction.

Findings

Tardif's formula for fractional chromatic number applies to the fractional Haemers bound when it equals the clique number.

Provides a general upper bound on the fractional Haemers bound after the Mycielski construction.

The bound is tight when the fractional Haemers bound equals the clique number.

Abstract

We investigate the effect of the generalized Mycielski construction $M_r(G)$ on the complementary fractional Haemers bound $\bar{\mathcal{H}}_f(G; \mathbb{F})$, a parameter that depends on a graph $G$ and a field $\mathbb{F}$. The effect of the Mycielski construction on graph parameters has already been studied for the fractional chromatic number $\chi_f$ and the complementary Lov\'asz theta number $\bar{\vartheta}$. Larsen, Propp, and Ullman provided a formula for $ \chi_f(M_2(G)) $ in terms of $\chi_f(G)$. This was later generalized by Tardif to $ \chi_f(M_r(G)) $ for any $r$, and Simonyi and the author gave a similar expression for $ \bar{\vartheta}(M_2(G)) $ in terms of $\bar{\vartheta}(G)$. In this paper, we show that Tardif's formula for the fractional chromatic number remains valid for $ \bar{\mathcal{H}}_f $ whenever $ \bar{\mathcal{H}}_f(G; \mathbb{F})$ equals the clique number of $G$. In particular, we provide a general upper bound on $\bar{\mathcal{H}}_f(M_r(G); \mathbb{F})$ in terms of $\bar{\mathcal{H}}_f(G;\mathbb{F})$ and we prove that this bound is tight whenever $ \bar{\mathcal{H}}_f(G; \mathbb{F})$ equals the clique number of $G$.