Fractional chromatic index

Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common.

Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems.

A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors. Equivalently, it can be defined as a homomorphism to the Kneser graph KG_a,b. The b-fold chromatic number χ_b(G) is the least a such that an a:b-coloring exists.

The fractional chromatic number χ_f(G) is defined to be

Note that the limit exists because χ_b(G) is subadditive, meaning χ_a+b(G) ≤ χ_a(G) + χ_b(G).

The fractional chromatic number can equivalently be defined in probabilistic terms. χ_f(G) is the smallest k for which there exists a probability distribution over the independent sets of G such that for each vertex v, given an independent set S drawn from the distribution,

Some properties of χ_f(G):

and

Furthermore, the fractional chromatic number approximates the chromatic number within a logarithmic factor, in fact:

Here n(G) is the order of G, α(G) is the independence number, ω(G) is the clique number, and χ(G) is the chromatic number. Kneser graphs give examples where χ(G)/χ_f(G) is arbitrarily large, since χ(KG_m,n)=m-2n+2, while χ_f(KG_m,n)=m/n.

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