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A zero-free interval for chromatic polynomials

1992
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(92)90614-l
fatcat:3xahu33rxvf7dahd4zrgsjeisi
*A**zero*-*free**interval**for**chromatic**polynomials*, Discrete Mathematics 101 (1992) 333-341. ...*For*example,*a*near-triangulation with all its n vertices in the bounding circuit has*chromatic**polynomial*t(t -l)(t -2)n-2, which is non-*zero*throughout the*interval*(2,2.5) but has the 'wrong' sign when ... And the graph of the octahedron minus an edge (H4,4 in Fig. 2 ) has*chromatic**polynomial*t(t -l)(t -2)(t3 -St2 + 23t -23), with*a**zero*at about 2.43. ...##
###
A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees
[article]

2015
*
arXiv
*
pre-print

It is proved that if G is

arXiv:1510.00417v1
fatcat:gujgijrksbdcxcqzzs4vlz66dm
*a*graph containing*a*spanning tree with at most three leaves, then the*chromatic**polynomial*of G has no roots in the*interval*(1,t_1], where t_1 ≈ 1.2904 is the smallest real ... We employ the Whitney 2-switch operation to manage the analysis of an infinite class of*chromatic**polynomials*. ...*A**Zero**Free**Interval**for*P (G i,j,k , t) We now determine the behaviour of the*chromatic*roots of each G i,j,k . ...##
###
A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

2011
*
SIAM Journal on Discrete Mathematics
*

Let G = (V, E) be

doi:10.1137/100790057
fatcat:7v257x7bejaihhw3yzqfyilrfm
*a*2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let P G (q) denote the*chromatic**polynomial*of G. ... which are not incident to*a*vertex of C, w e ∈ W 2*for*all e ∈ E(C), w e ∈ W 1*for*all other edges e, and W 1 , W 2 are suitably chosen*intervals*with −1 ∈ W 1 ⊂ W 2 ⊆ (−2, 0). ... Introduction The study of*chromatic**polynomials*of graphs was initiated by Birkhoff [3]*for*planar graphs in 1912 and,*for*general graphs, by Whitney [14, 15] in 1932. ...##
###
Chromatic zeros and generalized Fibonacci numbers

2009
*
Applicable Analysis and Discrete Mathematics
*

We prove that all 2n-anacci numbers and all their natural powers cannot be

doi:10.2298/aadm0902330a
fatcat:wdeoqwcoancb7jd5nyjrdsfhhi
*zeros*of any*chromatic**polynomial*. Also we investigate (2n + 1)-anacci numbers as*chromatic**zeros*. ... In this article we consider the problem whether generalized Fibonacci constants can be*zeros*of*chromatic**polynomials*. ... An*interval*is called*a**zero*-*free**interval**for**a**chromatic**polynomial*P (G, λ) if G has no*chromatic**zero*in this*interval*. ...##
###
A survey on the study of real zeros of flow polynomials

2019
*
Journal of Graph Theory
*

*For*

*a*bridgeless graph G, its flow

*polynomial*is defined to be the function F(G,q) which counts the number of nonwhere-

*zero*Γ-flows on an orientation of G whenever q is

*a*positive integer and Γ is an additive ... This article gives

*a*survey on the results and problems on the study of real

*zeros*of flow

*polynomials*. ...

*For*general graphs, (−∞, 0), (0, 1) and (1, 32/27] are the only maximal

*zero*-

*free*

*intervals*

*for*all

*chromatic*

*polynomials*, where an

*interval*is said to be

*zero*-

*free*

*for*

*a*function if it has no

*zero*in ...

##
###
Chromatic zeros and the golden ratio

2009
*
Applicable Analysis and Discrete Mathematics
*

An

doi:10.2298/aadm0901120a
fatcat:oaqfiyaoobbdtoi5rjdqwop3qe
*interval*is called*a*root-*free**interval**for**a**chromatic**polynomial*P (G, λ) if G has no*chromatic*root in this*interval*. ... It is well-known that (−∞, 0) and (0, 1) are two maximal root-*free**intervals**for*the family of all graphs (see [2] ). ...##
###
Zero-free intervals of chromatic polynomials of mixed hypergraphs
[article]

2020
*
arXiv
*
pre-print

In this paper, we prove that (-∞, 0) is

arXiv:1812.01814v2
fatcat:svrptivmtrcc3gdqdp37p2vusi
*a**zero*-*free**interval**for**chromatic**polynomials*of*a*family L_0 of hypergraphs and (0, 1) is*a**zero*-*free**interval**for**chromatic**polynomials*of*a*subfamily L_0' of ... These results extend known results on*zero*-*free**intervals*of*chromatic**polynomials*of graphs and hypergraphs. ... Acknowledgment: The authors wish to thank the referees*for*their very helpful comments and suggestions. The research was partially supported by NTU AcRF project (RP 3/16 DFM) of Singapore. ...##
###
Problems on chromatic polynomials of hypergraphs

2020
*
Electronic Journal of Graph Theory and Applications
*

*Chromatic*

*polynomials*of graphs have been studied extensively

*for*around one century. The concept of

*chromatic*

*polynomial*of

*a*hypergraph is

*a*natural extension of

*chromatic*

*polynomial*of

*a*graph. ... It also has been studied

*for*more than 30 years. This short article will focus on introducing some important open problems on

*chromatic*

*polynomials*of hypergraphs. ... Is there

*a*

*zero*-

*free*

*interval*

*for*

*chromatic*

*polynomials*of hypergraphs? In particular, is there

*a*

*zero*-

*free*

*interval*(

*a*, b) within some

*intervals*(−∞, 0) or (0, 1)? ...

##
###
Algebraic Integers as Chromatic and Domination Roots

2012
*
International Journal of Combinatorics
*

Since

doi:10.1155/2012/780765
fatcat:aa2lbozinrf67nhmszv7l3eeua
*chromatic**polynomial*and domination*polynomial*are monic*polynomial*with integer coefficients, its*zeros*are algebraic integers. ... This naturally raises the question: which algebraic integers can occur as*zeros*of*chromatic*and domination*polynomials*? In this paper, we state some properties of this kind of algebraic integers. ... An*interval*is called*a**zero*-*free**interval**for**a**chromatic*domination*polynomial*, if G has no*chromatic*domination*zero*in this*interval*. ...##
###
Graphs with chromatic roots in the interval (1,2)
[article]

2007
*
arXiv
*
pre-print

We present an infinite family of 3-connected non-bipartite graphs with

arXiv:0704.2264v1
fatcat:gpduofmhczavbjfi7h3wngv5um
*chromatic*roots in the*interval*(1,2) thus resolving*a*conjecture of Jackson's in the negative. ... In addition, we briefly consider other graph classes that are conjectured to have no*chromatic*roots in (1,2). ... What is the largest δ such that (1, δ) is*a**chromatic*-root-*free**interval**for*3connected non-bipartite graphs? Question 7. Which classes of graphs have no*chromatic*roots in (1, 2)? ...##
###
Orbital Chromatic and Flow Roots

2006
*
Combinatorics, probability & computing
*

We show,

doi:10.1017/s0963548306008200
fatcat:m3g3gyq2o5ejbl47zo7xzecxwu
*for*example, that they are dense in R, but under certain hypotheses, there are*zero*-*free*regions. ... The*chromatic**polynomial*P Γ (x) of*a*graph Γ is*a**polynomial*whose value at the positive integer k is the number of proper kcolourings of Γ. ... Parity and*zero*-*free**intervals*However, under parity conditions, we do get*zero*-*free**intervals**for*orbital*chromatic*roots. ...##
###
Graphs with Chromatic Roots in the Interval $(1,2)$

2007
*
Electronic Journal of Combinatorics
*

We present an infinite family of 3-connected non-bipartite graphs with

doi:10.37236/1019
fatcat:wbx3pjgmibgvvelbcky6fxsiom
*chromatic*roots in the*interval*$(1,2)$ thus resolving*a*conjecture of Jackson's in the negative. ... In addition, we briefly consider other graph classes that are conjectured to have no*chromatic*roots in $(1,2)$. ... However*for*certain classes of graphs the*chromatic*-root-*free**interval*can be extended -*for*example, Thomassen [6] showed that graphs with*a*hamiltonian path have no*chromatic*roots in (1, 1.29559 . ...##
###
On the real roots of σ-Polynomials
[article]

2016
*
arXiv
*
pre-print

It is known that the closure of the real roots of

arXiv:1611.09525v1
fatcat:qp5uyvuykfhnnkzh6g47hdq3zi
*chromatic**polynomials*is precisely {0, 1} [32/27,∞), with (-∞,0), (0,1) and (1,32/27) being maximal*zero*-*free**intervals**for*roots of*chromatic**polynomials*... We ask here whether such maximal*zero*-*free**intervals*exist*for*σ-*polynomials*, and show that the only such*interval*is [0,∞) -- that is, the closure of the real roots of σ-*polynomials*is (-∞,0]. ... Acknowledgments: This research was partially supported by*a*grant from NSERC. ...##
###
Page 6763 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

Summary: “The maximal

*zero*-*free**intervals**for**chromatic*poly- nomials of graphs are precisely (—00,0), (0,1), (1, 33). ...*For*example, the*zeros*of*chromatic**polynomials*of graphs of tree-width at most k consist of 0,1 and*a*dense subset of the*interval*( 34k hey W. T. ...##
###
Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

2022
*
Mathematics
*

In this article, we show that

doi:10.3390/math10020193
fatcat:b6awmfis3rhnni63axvwmu5lqy
*chromatic**polynomials*of mixed hypergraphs under certain conditions are*zero*-*free*in the*intervals*(−∞,0) and (0,1), which extends known results on*zero*-*free**intervals*of ...*chromatic**polynomials*of graphs and hypergraphs. ... result to*a*larger family of hypergraphs and proved that the existence of families of hypergraphs whose*chromatic**polynomials*are*zero*-*free*in the*intervals*(−∞, 0) and (0, 1). ...
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