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

Nabil Kahale, Leonard J. Schulman
1996 Combinatorica
A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method.  ...  An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian.  ...  Linial and R. Stanley for helpful consultations, and to N. Alon and an anonymous referee for several suggestions that improved the paper.  ...

### Bivariate Chromatic Polynomials of Mixed Graphs [article]

2021 arXiv   pre-print
The bivariate chromatic polynomial χ_G(x,y) of a graph G = (V, E), introduced by Dohmen-Pönitz-Tittmann (2003), counts all x-colorings of G such that adjacent vertices get different colors if they are  ...  Our main results is a decomposition formula which expresses χ_G(x,y) as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for χ_G  ...  for chromatic polynomials: (−1) |V | χ G (−x) equals the number of pairs of an x-coloring and a compatible acyclic orientation   ...

### Separating polynomial χ-boundedness from χ-boundedness [article]

Marcin Briański, James Davies, Bartosz Walczak
2022 arXiv   pre-print
chromatic number of a graph in 𝒢 with clique number n is equal to f(n) for every n∈ℕ.  ...  Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function fℕ→ℕ∪{∞} with f(1)=1 and f(n)≥3n+13, we construct a hereditary class of graphs 𝒢 such that the maximum  ...  See  and  for an explicit construction of the graphs G k with the appropriate acyclic orientations, based on Zykov's construction.  ...

### Efficient computation of the oriented chromatic number of recursively defined digraphs [article]

Frank Gurski and Dominique Komander and Marvin Lindemann
2021 arXiv   pre-print
We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.  ...  In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs.  ...  Acknowledgements The work of the second and third author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -388221852  ...

### The sandwich line graph

Denis Cornaz, Philippe Meurdesoif
2010 Electronic Notes in Discrete Mathematics
We observe that ω(G) + χ(S( G)) = n = ω(S( G)) + χ(G) for any graph G with n vertices, where G is any acyclic orientation of G and where S( G) is the (complement of the) auxiliary line graph introduced  ...  (Where as usual, ω and χ denote the clique number and the chromatic number.)  ...  of CPU time that can be used more efficiently by computing S( G) for a number of (randomly generated) acyclic orientations and selecting the best one.  ...

### Negative results on acyclic improper colorings

Pascal Ochem
2005 Discrete Mathematics & Theoretical Computer Science
International audience Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$.  ...  We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs.  ...  The acyclic chromatic number χ a (G) is the minimum number of colors needed in an acyclic proper coloring of the graph G.  ...

### Oriented graph coloring

Eric Sopena
2001 Discrete Mathematics
An oriented k-coloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset  ...  We survey the main results that have been obtained on oriented graph colorings.  ...  Oriented and acyclic chromatic numbers One of the ÿrst problems that have been considered in the framework of oriented colorings was to characterize the families of graphs having bounded oriented chromatic  ...

### Parameterized Mixed Graph Coloring

Peter Damaschke
2019 Journal of combinatorial optimization
As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm.  ...  Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts.  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ...

### On the oriented chromatic index of oriented graphs

Pascal Ochem, Alexandre Pinlou, Éric Sopena
2008 Journal of Graph Theory
We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree.  ...  We also consider lower and upper bounds of oriented chromatic number in terms of oriented chromatic index.  ...  Oriented vertex-colorings have been studied by several authors in the last decade and the problem of bounding the oriented chromatic number has been investigated for graphs with bounded acyclic chromatic  ...

### Page 47 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews
A brief survey on Read’s conjecture of unimodality of the coef- ficients of a chromatic polynomial is presented.  ...  The author investigates the occurrence, the upper bound on the number of up- sets in the teams, and the minimum number of upsets in the class of all irreducible bipartite tournaments, and obtains the number  ...

### Page 4112 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
The following lower bound is also established: If G is a graph with m edges whose chromatic number k is at least as large as its girth, then any acyclic orientation of G has at least m — 2m/k dependent  ...  Also, the fourth maximum chromatic polynomials P(G;A) for A= 3 in the class of 2-connected graphs of order n and all extremal graphs are deduced for every n > 5.”  ...

### Acyclic coloring of special digraphs [article]

Frank Gurski and Dominique Komander and Carolin Rehs
2020 arXiv   pre-print
An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets.  ...  We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width.  ...  Acknowledgements The work of the second and third author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -388221852  ...

### Computing Graph Polynomials on Graphs of Bounded Clique-Width [chapter]

J. A. Makowsky, Udi Rotics, Ilya Averbouch, Benny Godlin
2006 Lecture Notes in Computer Science
We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n f (k) ), where  ...  We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width.  ...  Dubrov and A. Matsliach for their active participation in our seminar. We are indebted to B. Courcelle for suggesting Proposition 3 and to P. Hlinȇný and M. Noy for making [GHN06] avaible.  ...

### Algorithms [chapter]

2011 Graph Coloring Problems
168 10.2 Polynomial Approximation 169 10.3 Even Chromatic Graphs 169 10.4 Grundy Number 170 10.5 Achromatic Number of a Tree 171 10.6 On-Line Coloring 172 10.7 Edge-Coloring Multigraphs 174  ...  Jaeger's Circular Flow Conjecture Berge's Strong Path Partition Conjecture Berge's Directed Path-Conjecture Minimal Orientations of Critical Graphs Alon-Tarsi Orientations and Chromatic Number Bibliography  ...

### Page 1558 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews
Let H be a graph with A vertices and chromatic number k.  ...  Let G be a graph with n > 2 vertices and at least one edge and define the polynomial Rg(k) of degree n—2 by Re(k) = Pg(k +1)/k(k +1), where Pg(k) is the chromatic polynomial of G.  ...
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