Planar Triangulations with Real Chromatic Roots Arbitrarily Close to 4

Royle, Gordon F.

doi:10.1007/s00026-008-0347-0

Cited by 11 publications

(35 citation statements)

References 19 publications

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“…And one might hope that a method which improves Conjecture 1 for k ≥ 5 might also be applicable for the interval from 4 to 5. In contrast to this, Royle [6] has shown that planar graphs have chromatic real root arbitrarily close to 4.…”

Section: Introductionmentioning

confidence: 95%

“…However, if G has no separating triangles we can improve the inequality (8), since none of the polynomials in equations (5), (6) are the zero polynomial. More precisely,…”

Section: The Chromatic Polynomialmentioning

confidence: 99%

“…For if there is a separating triangle we apply induction to its exterior and then its interior. And if there is no separating triangle we apply equations (5), (6). (All polynomials on the right hand sides of (5), (6) are non-zero because they are chromatic polynomials of graphs with no loops.…”

Section: The Chromatic Polynomialmentioning

confidence: 99%

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The number of colorings of planar graphs with no separating triangles

Thomassen¹

2017

Journal of Combinatorial Theory, Series B

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A classical result of Birkhoff and Lewis implies that every planar graph with n vertices has at least 15 · 2 n−1 distinct 5-vertex-colorings. Equality holds for planar triangulations with n−4 separating triangles. We show that, if a planar graph has no separating triangle, then it has at least (2 + 10 −12 ) n distinct 5-vertex-colorings. A similar result holds for k-colorings for each fixed k ≥ 5. Infinitely many planar graphs without separating triangles have less than 2.252 n distinct 5-vertexcolorings. As an auxiliary result we provide a complete description of the infinite 6-regular planar triangulations.

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Section: Introductionmentioning

confidence: 95%

“…However, if G has no separating triangles we can improve the inequality (8), since none of the polynomials in equations (5), (6) are the zero polynomial. More precisely,…”

Section: The Chromatic Polynomialmentioning

confidence: 99%

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The number of colorings of planar graphs with no separating triangles

Thomassen¹

2017

Journal of Combinatorial Theory, Series B

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“…Jacobsen et al [7] extended this to show that B 7 , B 8 , B 9 and B 10 are likewise accumulation points of real chromatic roots of plane triangulations (namely, m per × n free pieces of the triangular lattice of width m up to 12). Finally, Royle [9] has recently exhibited a family of plane triangulations with chromatic roots converging to 4. We know that the sequence B n has a limit 4.…”

Section: Some Questions and Remarksmentioning

confidence: 99%

“…If the answer to Beraha's question is positive, then there exist planar graphs whose chromatic roots are arbitrarily close to 4. Note that by exhibition of Royle [9], we have a family of plane triangulations with chromatic roots converging to 4. Of course, it is an open question which other numbers in the interval (32/27, 4) can be accumulation points of real chromatic roots of planar graphs.…”

Section: Some Questions and Remarksmentioning

confidence: 99%

Chromatic zeros and generalized Fibonacci numbers

Alikhani

Peng

2009

APPL ANAL DISCRETE M

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Graph Polynomials and Their Applications I: The Tutte Polynomial

Ellis-Monaghan

Merino

2010

Structural Analysis of Complex Networks

117

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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Planar Triangulations with Real Chromatic Roots Arbitrarily Close to 4

Abstract: We exhibit infinite families of planar triangulations with real chromatic roots arbitrarily close to 4, thus resolving a long-standing conjecture in the affirmative.

Cited by 11 publications

References 19 publications

The number of colorings of planar graphs with no separating triangles

The number of colorings of planar graphs with no separating triangles

Chromatic zeros and generalized Fibonacci numbers

Graph Polynomials and Their Applications I: The Tutte Polynomial

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