Cubical coloring -- fractional covering by cuts and semidefinite programming

TL;DR

This paper introduces a new graph invariant measuring fractional covering by cuts, explores its relation to other parameters, computes it for hypercube-based graphs, and provides a semidefinite programming approximation algorithm.

Contribution

It defines a novel graph invariant related to fractional cut coverings, analyzes its properties, and develops a polynomial-time approximation method using semidefinite programming.

Findings

The invariant is computed for hypercube-based graphs.

The parameter relates to chromatic number and bipartite density.

A polynomial-time approximation algorithm is developed.

Abstract

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr\"om, F\"arnqvist, Jonsson, and Thapper. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246--265]).