On Graphs Having No Chromatic Zeros in (1,2)

Dong, Fengming; Koh, Khee Meng

doi:10.1137/04061787x

Cited by 6 publications

(24 citation statements)

References 9 publications

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“…Such an edge‐cut

S

is also written as

S goodbreakinfix= (V_{1}, V_{2})

. For any non‐separable graph

G

F (G, q)

can be further factorized in any one of the following cases, due to Jackson (see also):

G goodbreakinfix- e

is separable for some edge

e

;

G

has a proper edge‐cut

S

with

2 goodbreakinfix\leq ∣ S ∣ goodbreakinfix\leq 3

, where an edge‐cut

S

is said to be proper if

G goodbreakinfix- S

has no isolated vertices. For any graph

G

and any two vertices

u

and

v

G

, let

G goodbreakinfix+ u v

denote the graph obtained by adding a new edge joining

u

and

v

.…”

Section: Basic Properties Of Flow Polynomialsmentioning

confidence: 99%

A survey on the study of real zeros of flow polynomials

Dong

2019

Journal of Graph Theory

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For a bridgeless graph G, its flow polynomial is defined to be the function F(G,q), which counts the number of nonwhere‐zero normalΓ‐flows on an orientation of G whenever q is a positive integer and normalΓ is an additive Abelian group of order q. It was introduced by Tutte in 1950, and the locations of zeros of this polynomial have been studied by many researchers. This paper gives a survey on the results and problems on the study of real zeros of flow polynomials.

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“…Such an edge‐cut

S

is also written as

S goodbreakinfix= (V_{1}, V_{2})

. For any non‐separable graph

G

F (G, q)

can be further factorized in any one of the following cases, due to Jackson (see also):

G goodbreakinfix- e

is separable for some edge

e

;

G

has a proper edge‐cut

S

with

2 goodbreakinfix\leq ∣ S ∣ goodbreakinfix\leq 3

, where an edge‐cut

S

is said to be proper if

G goodbreakinfix- S

has no isolated vertices. For any graph

G

and any two vertices

u

and

v

G

, let

G goodbreakinfix+ u v

denote the graph obtained by adding a new edge joining

u

and

v

.…”

Section: Basic Properties Of Flow Polynomialsmentioning

confidence: 99%

A survey on the study of real zeros of flow polynomials

Dong

2019

Journal of Graph Theory

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“…By Theorem 5 again, for any G ∈ R 0 − {L, Z 3 , K 4 }, G contains at least at least ⌈ 27k 11 − 27 22 ⌉ + 2µ(6 − k) ≥ 9 flow roots in the interval (1,2), where k = |W (G)| ≥ 3. However, as I know, no much research is conducted on counting the number of real flow roots of a graph in the interval (1,2), except some study which confirms certain families of graphs having no real flow roots in the interval (1, 2) (see [3,4,10,11,12]…”

Section: Proofmentioning

confidence: 99%

On Graphs whose Flow Polynomials have Real Roots Only

Dong

2018

Electron. J. Combin.

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Let G be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial F (G, λ) of G are integers if and only if G is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph G for which F (G, λ) has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for G is positive if and only if F (G, λ) does not have any real root in the interval (1, 2). We also prove that for any non-separable and 3-edge connected G, if G − e is also non-separable for each edge e in G and every 3-edge-cut of G consists of edges incident with some vertex of G, then all roots of P (G, λ) are real if and only if either G ∈ {L, Z 3 , K 4 } or F (G, λ) contains at least 9 real roots in the interval (1, 2), where L is the graph with one vertex and one loop and Z 3 is the graph with two vertices and three parallel edges joining these two vertices.Mathematics Subject Classifications: 05C21, 05C31 * This paper was partially supported by NTU AcRF project (RP 3/16 DFM) of Singapore.

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“…The graph X(s, t) is 3-connected and not bipartite and if s, t ≥ 3 are both odd, then it has a chromatic root in (1,2).…”

Section: An Infinite Family Of Counterexamplesmentioning

confidence: 99%

“…In this paper we demonstrate that this conjecture is false, by providing an infinite family of 3-connected non-bipartite graphs with chromatic roots in (1,2), and then briefly discuss some alternative conjectures.…”

Section: Introductionmentioning

confidence: 97%