On the Fractional Intersection Number of a Graph

Abstract

An intersection representation of a graph G is a function f : VOGU!2 S (where S is any set) with the property that uv A EOGU if and only if fOuUV fOvU0 h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that program gives the fractional intersection number. This is in consonance with fractional versions of other graph invariants such as matching number, chromatic number, edge chromatic number, etc. We examine cases where the fractional and ordinary intersection numbers are the same (interval and chordal graphs), as well as cases where they are wildly dierent (complete multipartite graphs). We find the fractional intersection number of almost all graphs by considering random graphs.