Fractional Cocoloring of Graphs

Abstract

The cochromatic number Z(G) of a graph G is the fewest number of colors needed to color the vertices of G so that each color class is a clique or an independent set. In a fractional cocoloring of G a non-negative weight is assigned to each clique and independent set so that for each vertex v, the sum of the weights of all cliques and independent sets containing v is at least one. The smallest total weight of such a fractional cocoloring of G is the fractional cochromatic number \(Z_f(G)\). In this paper we prove results for the fractional cochromatic number \(Z_f(G)\) that parallel results for Z(G) and the well studied fractional chromatic number \(\chi _f{(G)}\). For example \(Z_f(G)=\chi _f(G)\) when G is triangle-free, except when the only nontrivial component of G is a star. More generally, if G contains no k-clique, then \(Z_f(G)\le \chi _f(G)\le Z_f(G)+R(k,k)\), where R(k, k) is the minimum integer n such that every n-vertex graph has a k-clique or an independent set of size k. Moreover, every graph G with \(\chi _f(G)=m\) contains a subgraph H with \(Z_f(H)\ge (\frac{1}{4} - o(1))\frac{m}{\log _2 m}\). We also prove that the maximum value of \(Z_f(G)\) over all graphs G of order n is \(\varTheta (n/\log n)\), and the maximum over all graphs embedded on an orientable surface of genus g is \(\varTheta (\sqrt{g} / \log g)\).