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

Royle, Gordon F.

doi:10.37236/1019

Cited by 4 publications

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“…Jackson [7] conjectured that ω(S) = 2 for the family S of 3-connected nonbipartite graphs. However, counterexamples to this conjecture were discovered recently by Royle [10].…”

Section: Introductionmentioning

confidence: 99%

On Zero-Free Intervals in $(1,2)$ of Chromatic Polynomials of Some Families of Graphs

Dong¹,

Koh²

2010

SIAM J. Discrete Math.

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“…Jackson [7] conjectured that ω(S) = 2 for the family S of 3-connected nonbipartite graphs. However, counterexamples to this conjecture were discovered recently by Royle [10].…”

Section: Introductionmentioning

confidence: 99%

On Zero-Free Intervals in $(1,2)$ of Chromatic Polynomials of Some Families of Graphs

Dong¹,

Koh²

2010

SIAM J. Discrete Math.

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Graph Polynomials and Their Applications I: The Tutte Polynomial

Ellis-Monaghan

Merino

2010

Structural Analysis of Complex Networks

118

119

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We give a number of properties and combinatorial interpretations for various evaluations of the Tutte polynomial. We recover colorings, flows, orientations, network reliability, etc., and related polynomials as specializations of the Tutte polynomial. We discuss the coefficients, zeros, and derivatives of the Tutte polynomial, and conclude with a brief discussion of computational complexity. Preliminary NotionsThe graph terminology that we use is standard and generally follows Diestel [Die00]. Graphs may have loops and multiple edges. For a graph G we denote by V (G) its set of vertices and by E(G) its set of edges. An oriented graph, G, also called a digraph, has a direction assigned to each edge. Basic ConceptsWe first recall some of the notions of graph theory most used in this chapter. Two graphsWe denote by κ(G) the number of connected components of a graph G, and by c(G) the number of non-trivial connected components, that is the number of connected components not counting isolated vertices. A graph is k-connected if at least k vertices must be removed to disconnect the graph.A cycle in a graph G is a set of edges e 1 , . . . , e k such that, if e i = (v i , w i ) for 1 ≤ i ≤ k, then w i = v i+1 for 1 ≤ i ≤ k − 1; also w k = v 1 and v i = v j for i = j. A trail is a path that may revisit a vertex, but not retrace an edge. A circuit is a closed trail, and thus a cycle is just a circuit that does not revisit any vertices. In the case of a digraph, the edges of a trail or circuit must be consistently oriented.The dual notion of a cycle is that of cut or cocycle. If {V 1 , V 2 } is a partition of the vertex set, and the set C, consisting of those edges with one end in V 1 and one end in V 2 , is not empty, then C is called a cut. A cycle with one edge is called a loop and a cocycle with one edge is called a cut-edge or bridge. We refer to an edge that is neither a loop nor a bridge as ordinary.A tree is a connected graph without cycles. A forest is a graph whose connected components are all trees. A subgraph H of a graph G is spanning if V (H) = V (G). Spanning trees in connected graphs will play a fundamental role in the theory of the Tutte polynomial. Observe that a loop in a connected graph can be characterized as an edge that is in no spanning tree, while a bridge is an edge that is in every spanning tree.If V ⊆ V (G), then the induced subgraph on V has vertex set V and edge set those edges of G with both endpoints in V . If E ⊆ E(G), then the spanning subgraph induced by E has vertex set V (G) and edge set E .

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Domination numbers and zeros of chromatic polynomials

Dong

Koh

2008

Discrete Mathematics

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In this paper, we shall prove that if the domination number of G is at most 2, then P (G, ) is zero-free in the interval (1, ), where = 2 + 1 6 3 12 √ 93 − 108 − 1 6 3 12 √ 93 + 108 = 1.317672196 . . . , and P (G, ) = 0 for some graph G with domination number 2. We also show that if (G) v(G) − 2, then P (G, ) is zero-free in the interval (1, ), where = 5 3 + 1 6 3 12 √ 69 − 44 − 1 6 3 12 √ 69 + 44 = 1.430159709 . . . , and P (G, ) = 0 for some graph G with (G) = v(G) − 2.

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

Abstract: 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).

Cited by 4 publications

References 13 publications

On Zero-Free Intervals in $(1,2)$ of Chromatic Polynomials of Some Families of Graphs

On Zero-Free Intervals in $(1,2)$ of Chromatic Polynomials of Some Families of Graphs

Graph Polynomials and Their Applications I: The Tutte Polynomial

Domination numbers and zeros of chromatic polynomials

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