Cubical coloring — fractional covering by cuts and semidefinite programming

Robert Šámal

doi:10.46298/dmtcs.2134

Abstract

We introduce a new graph parameter 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 cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr&ouml;m, F&auml;rnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs &#x2013; properties and applications, DMTCS vol. 17:1, 2015, 33&#x2013;66]. 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.&nbsp;2, 246&#x2013;265]).

Highlights

All graphs we consider are undirected and loopless; to avoid trivialities we do not consider edgeless graphs
We introduce a new graph parameter that measures fractional covering of a graph by cuts
We provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number

Summary

Introduction

All graphs we consider are undirected and loopless; to avoid trivialities we do not consider edgeless graphs. There is an optimal solution y∗ of the above program, that respects symmetries of G: if there is an automorphism of G that maps edge e to edge f , y∗(e) = y∗(f ) This decreases the size of the linear program for graphs with nontrivial automorphism group.) in the final section we use this dual program to discuss yet another definition of x(G) in terms of the bipartite subgraph polytope. There is another possibility to dualize the notion of fractional cut covering, namely fractional cycle covering. Whether for some fixed k every cubic bridgeless graph has a cycle 3k/2k-cover

Basic properties

Cubical chromatic number of random graphs

Measuring the scale

Semidefinite approximation

Concluding Remarks