Chromatic numbers and a Lov\'asz type inequality for non-commutative graphs

TL;DR

This paper extends classical graph parameters like chromatic number and Lovász theta to non-commutative graphs, establishing bounds and generalizations of key theorems in this operator space framework.

Contribution

It introduces two new non-commutative chromatic numbers and generalizes the Lovász sandwich inequality, Sabadussi's Theorem, and Hedetniemi's conjecture.

Findings

Chromatic number of the orthogonal complement is bounded below by the theta number.

Generalization of Lovász sandwich inequality to non-commutative graphs.

Extension of classical graph theorems to the non-commutative setting.

Abstract

Non-commutative graph theory is an operator space generalization of graph theory. Well known graph parameters such as the independence number and Lov\'asz theta function were first generalized to this setting by Duan, Severini, and Winter. We introduce two new generalizations of the chromatic number to non-commutative graphs and provide a generalization of the Lov\'asz sandwich inequality. In particular, we show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabadussi's Theorem and Hedetniemi's conjecture to non-commutative graphs.