The Go polynomials of a graph

Abstract

This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This definition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less “local” in nature than those used to define more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modifications similar in spirit to the deletion–contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also briefly record some correlation inequalities.