Cubical coloring — fractional covering by cuts

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. As a main result, we compute the parameter for infinitely many 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 Engström, Färnqvist, Jonsson, and Thapper. This paper is an extension of extended abstract, that appeared as [20]. Another previous treatment of this topics appears in the author’s thesis [21].