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### A zero-free interval for chromatic polynomials

D.R. Woodall
1992 Discrete Mathematics
., 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]

Thomas Perrett
2015 arXiv   pre-print
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 .  ...

### A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

F. M. Dong, Bill Jackson
2011 SIAM Journal on Discrete Mathematics
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.  ...

### Chromatic zeros and generalized Fibonacci numbers

Saeid Alikhani, Yee-Hock Peng
2009 Applicable Analysis and Discrete Mathematics
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.  ...

### A survey on the study of real zeros of flow polynomials

Fengming Dong
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

Saeid Alikhani, Yee-Hock Peng
2009 Applicable Analysis and Discrete Mathematics
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] ).  ...

### Zero-free intervals of chromatic polynomials of mixed hypergraphs [article]

Ruixue Zhang, Fengming Dong
2020 arXiv   pre-print
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.  ...

### Problems on chromatic polynomials of hypergraphs

Ruixue Zhang, Fengming Dong
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

Saeid Alikhani, Roslan Hasni
2012 International Journal of Combinatorics
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.  ...

### Graphs with chromatic roots in the interval (1,2) [article]

Gordon F. Royle
2007 arXiv   pre-print
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)?  ...

### Orbital Chromatic and Flow Roots

PETER J. CAMERON, K. K. KAYIBI
2006 Combinatorics, probability & computing
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.  ...

### Graphs with Chromatic Roots in the Interval \$(1,2)\$

Gordon F. Royle
2007 Electronic Journal of Combinatorics
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 .  ...

### On the real roots of σ-Polynomials [article]

Jason Brown, Aysel Erey
2016 arXiv   pre-print
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.  ...

### 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

Ruixue Zhang, Fengming Dong, Meiqiao Zhang
2022 Mathematics
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).  ...
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