Dual Hoffman Bounds for the Stability and Chromatic Numbers Based on SDP

TL;DR

This paper introduces dual Hoffman bounds derived from SDP norms that connect the stability and chromatic numbers of graphs, unifying and strengthening existing bounds through duality and convex optimization techniques.

Contribution

It presents a new family of norms providing dual bounds for stability and chromatic numbers, linking Hoffman's and Lovász theta bounds via duality and strengthening previous convex bounds.

Findings

New norms achieve Lovász theta function at their optimum.

Hoffman's bound and Delsarte-Hoffman ratio bound are formally dual.

Weighted bounds relate stability and fractional chromatic numbers.

Abstract

The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs, and is further and deeply developed in the context of the Lov\'asz theta function and its equivalent characterizations and variants. The main achievement of this paper is the introduction of a new family of norms, providing upper bounds for the stability number, that are obtained from duality from the norms motivated by Hoffman's lower bound for the chromatic number and which achieve the (complementary) Lov\'asz theta function at their optimum. As a consequence, our norms make it formal that Hoffman's bound for the chromatic number and the Delsarte-Hoffman ratio bound for the stability number are indeed dual. Further, we show that our new bounds strengthen the convex quadratic bounds for the stability number studied by Luz and Schrijver, and which achieve the Lov\'asz theta function at their optimum. One of the key observations regarding weighted versions of these bounds is that, for any upper bound for the stability number of a graph which is a positive definite monotone gauge function, its gauge dual is a lower bound on the fractional chromatic number, and conversely. Our presentation is elementary and accessible to a wide audience.