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

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.

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Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph

COMBINATORICA ◽

10.1007/bf01261322 ◽

1996 ◽

Vol 16 (3) ◽

pp. 383-397 ◽

Cited By ~ 7

Author(s):

Nabil Kahale ◽

Leonard J. Schulman

Keyword(s):

Chromatic Polynomial ◽

Acyclic Orientations

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Network reliability and acyclic orientations

Networks ◽

10.1002/net.3230140402 ◽

1984 ◽

Vol 14 (4) ◽

pp. 489-505 ◽

Cited By ~ 22

Author(s):

R. Johnson

Keyword(s):

Network Reliability ◽

Acyclic Orientations

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Some Alternate Characterizations of Reliability Domination

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800001571 ◽

1990 ◽

Vol 4 (2) ◽

pp. 257-276 ◽

Cited By ~ 5

Author(s):

F. T. Boesch ◽

A. Satyanarayana ◽

C. L. Suffel

Keyword(s):

Spanning Trees ◽

Reliability Theory ◽

Chromatic Polynomial ◽

Acyclic Orientations ◽

Graph Theoretic ◽

System A ◽

Finite Set ◽

Special Case

An important problem in reliability theory is to determine the reliability of a system from the reliability of its components. If E is a finite set of components, then certain subsets of E are prescribed to be the operating states of the system. A formation is any collection F of minimal operating states whose union is E. Reliability domination is defined as the total number of odd cardinality formations minus the total number of even cardinality formations. The purpose of this paper is to establish some new results concerning reliability domination. In the special case where the system can be identified with a graph or digraph, these new results lead to some new graph-theoretic properties and to simple proofs of certain known theorems. The pertinent graph-theoretic properties include spanning trees, acyclic orientations, Whitney's broken cycles, and Tutte's internal activity associated with the chromatic polynomial.

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