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

A generalized chromatic polynomial, acyclic orientations with prescribed sources and sinks, and network reliability

AbstractSuppose G=(V,E) is a graph and K, K′, K″ are subsets of V such that K⊆K′ ∩ K″. We introduce and study a polynomial P(G,K,K′,K″; λ) in λ. This polynomial coincides with the classical chromatic polynomial P(G; λ) when K=V. The results of this paper generalize Whitney's characterizations of the coefficients of P(G; λ) and the work of Stanley on acyclic orientations. Furthermore, we establish a connection between a family of polynomials associated with network reliability and a family of polynomials associated with P(G,K,K′,K″; λ)

Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes

Tubes Klaus Dohmen Humboldt-Universitat zu Berlin Institut fur Informatik Unter den Linden 6 D-10099 Berlin, Germany e-mail: [email protected] Abstract Recently, Naiman and Wynn introduced the concept of an abstract tube in order to obtain improved inclusion-exclusion identities and inequalities that involve much fewer terms than their classical counterparts. In this paper, we introduce a particular class of abstract tubes which plays an important role with respect to chromatic polynomials and network reliability. The inclusionexclusion identities and inequalities associated with this class simultaneously generalize several wellknown results such as Whitney's broken circuit theorem, Shier's expression for the reliability of a network as an alternating sum over chains in a semilattice and Narushima's inclusion-exclusion identity for posets. Moreover, we show that under some restrictive assumptions a polynomial time inclusion-exclusion algorithm can be devised, which gener..