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

COMBINATORICA ◽  
1996 ◽  
Vol 16 (3) ◽  
pp. 383-397 ◽  
Author(s):  
Nabil Kahale ◽  
Leonard J. Schulman
Keyword(s):  
Chromatic Polynomial ◽  
Acyclic Orientations

scholarly journals A Quasisymmetric Function Generalization of the Chromatic Symmetric Function

10.37236/518 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Brandon Humpert
Keyword(s):  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Quasisymmetric Function ◽  
Quasisymmetric Functions ◽  
Acyclic Orientations ◽  
Chromatic Symmetric Function

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.

Some Alternate Characterizations of Reliability Domination

1990 ◽  
Vol 4 (2) ◽  
pp. 257-276 ◽  
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.

scholarly journals Counting and Sampling Markov Equivalent Directed Acyclic Graphs

2019 ◽  
Vol 33 ◽  
pp. 7984-7991
Author(s):  
Topi Talvitie ◽  
Mikko Koivisto
Keyword(s):  
Equivalence Class ◽  
Computational Cost ◽  
Chordal Graph ◽  
Directed Acyclic Graphs ◽  
Desktop Computer ◽  
Bounded Degree ◽  
Acyclic Orientations ◽  
Uniform Sampling ◽  
Acyclic Graphs ◽  
Graph Size

Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2nn4) arithmetic operations, improving a previous superexponential upper bound. The second requires O(k!2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals A result on co-chromatic graphs

1981 ◽  
Vol 4 (2) ◽  
pp. 365-369 ◽  
Author(s):  
E. J. Farrell
Keyword(s):  
Chromatic Polynomial ◽  
Sufficient Condition

A sufficient condition for two graphs with the same number of nodes to have the same chromatic polynomial is given.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

scholarly journals An extension of the bivariate chromatic polynomial

2010 ◽  
Vol 31 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Ilia Averbouch ◽  
Benny Godlin ◽  
J.A. Makowsky
Keyword(s):  
Chromatic Polynomial
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