New bounds for the max-$k$-cut and chromatic number of a graph

TL;DR

This paper introduces new eigenvalue-based bounds for the max-$k$-cut and chromatic number of graphs using semidefinite programming relaxations, applicable to various graph classes and extending previous bounds.

Contribution

It presents the first eigenvalue bound for max-$k$-cut applicable to any graph and derives a new eigenvalue bound on the chromatic number, analyzing their tightness and applicability.

Findings

The weakest SDP relaxation yields a closed-form eigenvalue bound involving the largest Laplacian eigenvalue.

The eigenvalue bound for max-$k$-cut is tight for certain classes of graphs.

The new chromatic number bound matches Hoffman bound for regular graphs but differs in general.

Abstract

We consider several semidefinite programming relaxations for the max-$k$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general. We prove that the eigenvalue bound for the max-$k$-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme.