Chromatic polynomials and network reliability

AbstractIn this paper, we introduce and study an extension of the chromatic polynomial of a graph. The new polynomial, determined by a graph G and a subset K of points of G, coincides with the classical chromatic polynomial when K is the set of all points of G. The main theorems in the present paper include analogues of the standard axiomatic characterization and Whitney's topological characterizations of the chromatic polynomial, and the theorem of Stanley relating the chromatic polynomial to the number of acylic of G. The work in this paper was stimulated by important connections between the chromatic polynomial and the all-terminal network reliability problem, and by recent work of Boesch, Satyanarayana, and Suffel on a graph invariant related to the K-terminal reliability problem. Several of the Boesch, Satyanarayana, and Suffel are derived as corollaries to the main theorems of the present paper