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A zero-free interval for chromatic polynomials
1992
Discrete Mathematics
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., 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. ...
doi:10.1016/0012-365x(92)90614-l
fatcat:3xahu33rxvf7dahd4zrgsjeisi
A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees
[article]
2015
arXiv
pre-print
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It is proved that if G is 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 . ...
arXiv:1510.00417v1
fatcat:gujgijrksbdcxcqzzs4vlz66dm
A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs
2011
SIAM Journal on Discrete Mathematics
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Let G = (V, E) be 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. ...
doi:10.1137/100790057
fatcat:7v257x7bejaihhw3yzqfyilrfm
Chromatic zeros and generalized Fibonacci numbers
2009
Applicable Analysis and Discrete Mathematics
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We prove that all 2n-anacci numbers and all their natural powers cannot be 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. ...
doi:10.2298/aadm0902330a
fatcat:wdeoqwcoancb7jd5nyjrdsfhhi
A survey on the study of real zeros of flow polynomials
2019
Journal of Graph Theory
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2020 pre-print arXiv:2007.05195v1 fatcat:7exdsrz3tzdjxf4adnmmkqrao4
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 ...
doi:10.1002/jgt.22458
fatcat:j4jrgxgvrjf6hc5kcgpwkp2odq
Chromatic zeros and the golden ratio
2009
Applicable Analysis and Discrete Mathematics
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An 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] ). ...
doi:10.2298/aadm0901120a
fatcat:oaqfiyaoobbdtoi5rjdqwop3qe
Zero-free intervals of chromatic polynomials of mixed hypergraphs
[article]
2020
arXiv
pre-print
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In this paper, we prove that (-∞, 0) is 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. ...
arXiv:1812.01814v2
fatcat:svrptivmtrcc3gdqdp37p2vusi
Problems on chromatic polynomials of hypergraphs
2020
Electronic Journal of Graph Theory and Applications
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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)? ...
doi:10.5614/ejgta.2020.8.2.4
fatcat:b6yzriyz3fbwlh6j4zzkntw4k4
Algebraic Integers as Chromatic and Domination Roots
2012
International Journal of Combinatorics
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Since 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. ...
doi:10.1155/2012/780765
fatcat:aa2lbozinrf67nhmszv7l3eeua
Graphs with chromatic roots in the interval (1,2)
[article]
2007
arXiv
pre-print
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We present an infinite family of 3-connected non-bipartite graphs with 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)? ...
arXiv:0704.2264v1
fatcat:gpduofmhczavbjfi7h3wngv5um
Orbital Chromatic and Flow Roots
2006
Combinatorics, probability & computing
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We show, 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. ...
doi:10.1017/s0963548306008200
fatcat:m3g3gyq2o5ejbl47zo7xzecxwu
Graphs with Chromatic Roots in the Interval $(1,2)$
2007
Electronic Journal of Combinatorics
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We present an infinite family of 3-connected non-bipartite graphs with 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 . ...
doi:10.37236/1019
fatcat:wbx3pjgmibgvvelbcky6fxsiom
On the real roots of σ-Polynomials
[article]
2016
arXiv
pre-print
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It is known that the closure of the real roots of 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. ...
arXiv:1611.09525v1
fatcat:qp5uyvuykfhnnkzh6g47hdq3zi
Page 6763 of Mathematical Reviews Vol. , Issue 98K
[page]
1998
Mathematical Reviews
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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
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In this article, we show that 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). ...
doi:10.3390/math10020193
fatcat:b6awmfis3rhnni63axvwmu5lqy
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