scholarly journals Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations

2020 ◽  
Author(s):  
Fengming Dong ◽  
Jun Ge ◽  
Helin Gong ◽  
Bo Ning ◽  
Zhangdong Ouyang ◽  
...  
Keyword(s):  
Chromatic Polynomials ◽  
Acyclic Orientations

scholarly journals Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

10.37236/2741 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Matthias Beck ◽  
Tristram Bogart ◽  
Tu Pham
Keyword(s):  
Small Neighborhood ◽  
Fixed Number ◽  
Reciprocity Theorem ◽  
Additive Number Theory ◽  
Chromatic Polynomials ◽  
Acyclic Orientations ◽  
Research Activities ◽  
Golomb Rulers ◽  
Combinatorial Reciprocity Theorem ◽  
Mixed Graphs

A Golomb ruler is a sequence of distinct integers (the markings of the ruler) whose pairwise differences are distinct. Golomb rulers, also known as Sidon sets and $B_2$ sets, can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for optimal Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study$$g_m(t) := \# \left\{ {\bf x} \in {\bf Z}^{m+1} : \, 0 = x_0 < x_1 < \dots < x_m = t , \text{ all } x_j - x_k \text{ distinct} \right\} ,$$the number of Golomb rulers with $m+1$ markings and length $t$.Our main result is that $g_m(t)$ is a quasipolynomial in $t$ which satisfies a combinatorial reciprocity theorem: $(-1)^{m-1} g_m(-t)$ equals the number of rulers ${\bf x}$ of length $t$ with $m+1$ markings, each counted with its Golomb multiplicity, which measures how many combinatorially different Golomb rulers are in a small neighborhood of ${\bf x}$. Our reciprocity theorem can be interpreted in terms of certain mixed graphs associated to Golomb rulers; in this language, it is reminiscent of Stanley's reciprocity theorem for chromatic polynomials. Thus in the second part of the paper we develop an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.

scholarly journals Approximations for chromatic polynomials

1976 ◽  
Vol 20 (1) ◽  
pp. 5-19 ◽  
Author(s):  
N.L Biggs ◽  
G.H.J Meredith
Keyword(s):  
Chromatic Polynomials

scholarly journals On Weak Chromatic Polynomials of Mixed Graphs

2013 ◽  
Vol 31 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Matthias Beck ◽  
Daniel Blado ◽  
Joseph Crawford ◽  
Taïna Jean-Louis ◽  
Michael Young
Keyword(s):  
Chromatic Polynomials ◽  
Mixed Graphs

scholarly journals Note on chromatic polynomials of the threshold graphs

2019 ◽  
Vol 7 (2) ◽  
pp. 217-224
Author(s):  
Noureddine Chikh ◽  
◽  
Miloud Mihoubi ◽  
Keyword(s):  
Chromatic Polynomials ◽  
Threshold Graphs

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 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.

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