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 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 Chromatic Polynomials

10.37236/163 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Kerri Morgan ◽  
Graham Farr
Keyword(s):  
Upper Bound ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Separable Graph

The chromatic polynomial gives the number of proper $\lambda$-colourings of a graph $G$. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if $P({G},\lambda)=P({H_{1}},\lambda)P({H_{2}},\lambda)/P({K_{r}},\lambda)$ for some graphs $H_{1}$ and $H_{2}$ and clique $K_{r}$. It is known that the chromatic polynomial of any clique-separable graph, that is, a graph containing a separating $r$-clique, has a chromatic factorisation. We show that there exist other chromatic polynomials that have chromatic factorisations but are not the chromatic polynomial of any clique-separable graph and identify all such chromatic polynomials of degree at most 10. We introduce the notion of a certificate of factorisation, that is, a sequence of algebraic transformations based on identities for the chromatic polynomial that explains the factorisations for a graph. We find an upper bound of $n^{2}2^{n^{2}/2}$ for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order $\leq 9$.

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.

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 Chromatic Polynomials of Oriented Graphs

10.37236/8240 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Danielle Cox ◽  
Christopher Duffy
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Oriented Graph ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Interval Graphs ◽  
Chromatic Polynomials ◽  
Oriented Graphs ◽  
Real Roots ◽  
The Family

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying simple graph, closing an open problem posed by Sopena. We find that such oriented graphs can be both identified and constructed in polynomial time as they are exactly the family of quasi-transitive oriented co-interval graphs. We study the analytic properties of this polynomial and show that there exist oriented graphs which have chromatic polynomials have roots, including negative real roots,  that cannot be realized as the root of any chromatic polynomial of a simple graph.

Extended Chromatic Polynomials

1972 ◽  
Vol 24 (3) ◽  
pp. 492-501 ◽  
Author(s):  
Andrew Sobczyk ◽  
James O. Gettys
Keyword(s):  
Positive Integer ◽  
Finite Graph ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Vertex Set

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.

scholarly journals An Elementary Chromatic Reduction for Gain Graphs and Special Hyperplane Arrangements

10.37236/210 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Pascal Berthomé ◽  
Raul Cordovil ◽  
David Forge ◽  
Véronique Ventos ◽  
Thomas Zaslavsky
Keyword(s):  
Chromatic Polynomial ◽  
Hyperplane Arrangements ◽  
Chromatic Polynomials ◽  
Counting Functions ◽  
Group A ◽  
Universal Domain ◽  
Gain Graphs ◽  
Vertex Partitions ◽  
Gain Graph ◽  
Integral Gain

A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are analogous to those of the chromatic polynomial of an ordinary graph, though they are different from those usually assumed of gain graphs or matroids. The three laws lead to the weak chromatic group of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for all switching-invariant functions of gain graphs, such as chromatic polynomials, that satisfy the deletion-contraction identity for neutral links and are zero on graphs with neutral loops. Examples are the total chromatic polynomial of any gain graph, including its specialization the zero-free chromatic polynomial, and the integral and modular chromatic functions of an integral gain graph. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. The proof involves gain graphs between the Catalan and Shi graphs whose polynomials are expressed in terms of descending-path vertex partitions of the graph of $(-1)$-gain edges. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.

scholarly journals On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

10.37236/647 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jacob A. White
Keyword(s):  
Chromatic Polynomial ◽  
Partition Lattice ◽  
Chromatic Polynomials ◽  
Möbius Inversion

In this paper we introduce multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, Pönitz and Tittman. We prove that specializations of these new polynomials recover polynomials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs, all weighted according to certain statistics. We also prove that the polynomials can be defined in terms of Möbius inversion on the partition lattice of a hypergraph, and we compute these polynomials for various classes of hypergraphs. We also consider trivariate polynomials, which we call the hyperedge elimination polynomial and the trivariate chromatic polynomial.

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