scholarly journals On chordal graphs and their chromatic polynomials

2003 ◽  
Vol 93 (2) ◽  
pp. 240 ◽  
Author(s):  
Geir Agnarsson
Keyword(s):  
Explicit Formula ◽  
Chromatic Polynomial ◽  
Chordal Graphs ◽  
Chromatic Polynomials ◽  
Arbitrary Power ◽  
Triangulated Graph

We derive a formula for the chromatic polynomial of a chordal or a triangulated graph in terms of its maximal cliques. As a corollary we obtain a way to write down an explicit formula for the chromatic polynomial for an arbitrary power of a graph which belongs to any given class of chordal graphs that are closed under taking powers.

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

scholarly journals Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

2008 ◽  
Vol 17 (2) ◽  
pp. 225-238 ◽  
Author(s):  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Maximal Degree ◽  
Bounded Degree ◽  
Chromatic Polynomials ◽  
Image Position ◽  
Complex Zeros

We prove that the chromatic polynomial$P_\mathbb{G}(q)$of a finite graph$\mathbb{G}$of maximal degree Δ is free of zeros for |q| ≥C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

scholarly journals The 224 non-chordal graphs on less than 10 vertices whose chromatic polynomials have no complex roots

2001 ◽  
Vol 226 (1-3) ◽  
pp. 387-396 ◽  
Author(s):  
Ottavio M. D'Antona ◽  
Carlo Mereghetti ◽  
Fabio Zamparini
Keyword(s):  
Chordal Graphs ◽  
Chromatic Polynomials ◽  
Complex Roots

Linear layouts of weakly triangulated graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650038 ◽  
Author(s):  
Asish Mukhopadhyay ◽  
S. V. Rao ◽  
Sidharth Pardeshi ◽  
Srinivas Gundlapalli
Keyword(s):  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Interesting Connection ◽  
Triangulated Graphs ◽  
Distance Data ◽  
Point Placement ◽  
Appl Math ◽  
Block Tree ◽  
Triangulated Graph

A graph [Formula: see text] is said to be triangulated if it has no chordless cycles of length 4 or more. Such a graph is said to be rigid if, for a valid assignment of edge lengths, it has a unique linear layout and non-rigid otherwise. Damaschke [Point placement on the line by distance data, Discrete Appl. Math. 127(1) (2003) 53–62] showed how to compute all linear layouts of a triangulated graph, for a valid assignment of lengths to the edges of [Formula: see text]. In this paper, we extend this result to weakly triangulated graphs, resolving an open problem. A weakly triangulated graph can be constructively characterized by a peripheral ordering of its edges. The main contribution of this paper is to exploit such an edge order to identify the rigid and non-rigid components of [Formula: see text]. We first show that a weakly triangulated graph without articulation points has at most [Formula: see text] different linear layouts, where [Formula: see text] is the number of quadrilaterals (4-cycles) in [Formula: see text]. When [Formula: see text] has articulation points, the number of linear layouts is at most [Formula: see text], where [Formula: see text] is the number of nodes in the block tree of [Formula: see text] and [Formula: see text] is the total number of quadrilaterals over all the blocks. Finally, we propose an algorithm for computing a peripheral edge order of [Formula: see text] by exploiting an interesting connection between this problem and the problem of identifying a two-pair in [Formula: see text]. Using an [Formula: see text] time solution for the latter problem, we propose an [Formula: see text] time algorithm for computing its peripheral edge order, where [Formula: see text] and [Formula: see text] are respectively the number of edges and vertices of [Formula: see text]. For sparse graphs, the time complexity can be improved to [Formula: see text], using the concept of handles [R. B. Hayward, J. P. Spinrad and R. Sritharan, Improved algorithms for weakly chordal graphs, ACM Trans. Algorithms 3(2) (2007) 19pp].

Bounds For The Real Zeros of Chromatic Polynomials

2008 ◽  
Vol 17 (6) ◽  
pp. 749-759 ◽  
Author(s):  
F. M. DONG ◽  
K. M. KOH
Keyword(s):  
Maximum Degree ◽  
Short Proof ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Zeros ◽  
Complex Zeros ◽  
Special Case

Sokal in 2001 proved that the complex zeros of the chromatic polynomialPG(q) of any graphGlie in the disc |q| < 7.963907Δ, where Δ is the maximum degree ofG. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fernández and Procacci recently improved Sokal's result to |q| < 6.91Δ. In this paper, we shall show that all real zeros ofPG(q) are in the interval [0,5.664Δ). For the special case that Δ = 3, all real zeros ofPG(q) are in the interval [0,4.765Δ).

On injective chromatic polynomials of graphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550035
Author(s):  
Anjaly Kishore ◽  
M. S. Sunitha
Keyword(s):  
Chromatic Number ◽  
Discrete Math ◽  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Common Neighbor ◽  
Chromatic Polynomials ◽  
Wheel Graphs ◽  
Minimum Number

The injective chromatic number χi(G) [G. Hahn, J. Kratochvil, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256(1–2) (2002) 179–192] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. The nature of the coefficients of injective chromatic polynomials of complete graphs, wheel graphs and cycles is studied. Injective chromatic polynomial on operations like union, join, product and corona of graphs is obtained.

scholarly journals Galois groups of chromatic polynomials

2012 ◽  
Vol 15 ◽  
pp. 281-307 ◽  
Author(s):  
Kerri Morgan
Keyword(s):  
Galois Group ◽  
Systematic Study ◽  
Statistical Physics ◽  
Infinite Family ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Chromatic Polynomials ◽  
Algebraic Properties

AbstractThe chromatic polynomialP(G,λ) gives the number of ways a graphGcan be properly coloured in at mostλcolours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separableθ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.

scholarly journals Certificates of Factorisation for a Class of Triangle-Free Graphs

10.37236/164 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Kerri Morgan ◽  
Graham Farr
Keyword(s):  
Infinite Family ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Odd Cycle ◽  
Separable Graph

The chromatic polynomial $P(G,\lambda)$ gives the number of $\lambda$-colourings of a graph. If $P(G,\lambda)=P(H_{1},\lambda)P(H_{2},\lambda)/P(K_{r},\lambda)$, then the graph $G$ is said to have a chromatic factorisation with chromatic factors $H_{1}$ and $H_{2}$. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle $C_{2n+1}$, $n\geq2$, as a chromatic factor.

scholarly journals On Hyper-Chordal graphs

2021 ◽  
Vol 37 (1) ◽  
pp. 119-126
Author(s):  
MIHAI TALMACIU
Keyword(s):  
Combinatorial Optimization ◽  
Linear Complexity ◽  
Optimization Algorithms ◽  
Recognition Algorithm ◽  
Chordal Graphs ◽  
Induced Subgraph ◽  
Recognition Algorithms ◽  
Triangulated Graphs ◽  
Triangulated Graph

Triangulated graphs have many interesting properties (perfection, recognition algorithms, combinatorial optimization algorithms with linear complexity). Hyper-triangulated graphs are those where each induced subgraph has a hyper-simplicial vertex. In this paper we give the characterizations of hyper-triangulated graphs using an ordering of vertices and the weak decomposition. We also offer a recognition algorithm for the hyper-triangulated graphs, the inclusions between the triangulated graphs generalizations and we show that any hyper-triangulated graph is perfect.

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