Exponentially many 4‐list‐colorings of triangle‐free graphs on surfaces

Kelly, Tom; Postle, Luke

doi:10.1002/jgt.22153

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Let F be as stated, let G ∈ F , and let ξ : S 1 → Σ be a closed curve that bounds an open disk ∆ and intersects G only in vertices and ∆ includes a vertex of G. Then by Theorem 2.14 and [46,Theorem 7] the number of times, N, the curve ξ intersects G, counting multiplicities, satisfies N ≥ 3. By [46,Theorem 8] the number of vertices in ∆ plus N is at most 46N. Thus the number of vertices in ∆ is at most 45N ≤ 67.5(N − 1), as desired.…”

Section: Examples Pertaining To Exponentially Many Coloringsmentioning

confidence: 99%

Hyperbolic families and coloring graphs on surfaces

Postle

Thomas

2018

Trans. Amer. Math. Soc. Ser. B

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We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let G be a graph embedded in a fixed surface Σ of genus g and let L = (L(v) : v ∈ V (G)) be a collection of lists such that either each list has size at least five, or each list has size at least four and G is triangle-free, or each list has size at least three and G has no cycle of length four or less. An L-coloring of G is a mapping φ with domain V (G) such that φ(v) ∈ L(v) for every v ∈ V (G) and φ(v) = φ(u) for every pair of adjacent vertices u, v ∈ V (G). We prove• if every non-null-homotopic cycle in G has length Ω(log g), then G has an L-coloring,• if G does not have an L-coloring, but every proper subgraph does ("L-critical graph"), then |V (G)| = O(g), 2All graphs in this paper are finite, and have no loops or multiple edges. Our terminology is standard, and may be found in [11] or [18]. In particular, cycles and paths have no repeated vertices, and by a coloring of a graph G we mean a proper vertex-coloring; that is, a mapping φ with domain V (G) such that φ(u) = φ(v) whenever u, v are adjacent vertices of G. By a surface we mean a (possibly disconnected) compact 2-dimensional manifold with possibly empty boundary. The boundary of a surface Σ will be denoted by bd(Σ). By the classification theorem every surface Σ is obtained from the disjoint union of finitely many spheres by adding a handles, b crosscaps and removing the interiors of finitely many pairwise disjoint closed disks. In that case the Euler genus of Σ is 2a + b. Motivated by graph coloring problems we study families of graphs that satisfy the following isoperimetric inequalities. By an embedded graph we mean a pair (G, Σ), where Σ is a surface without boundary and G is a graph embedded in Σ. We will usually suppress the surface and speak about an embedded graph G, and when we will need to refer to the surface we will do so by saying that G is embedded in a surface Σ. Let F be a family of non-null embedded graphs. We say that F is hyperbolic if there exists a constant c > 0 such that if G ∈ F is a graph that is embedded in a surface Σ, then for every closed curve ξ : S 1 → Σ that bounds an open disk ∆ and intersects G only in vertices, if ∆ includes a vertex of G,The hyperbolic families of interest to us arise from coloring problems, more specifically from a generalization of the classical notion of coloring, introduced by Erdös, Rubin and Taylor [32] and known as list coloring or choosability. We need a few definitions in order to introduce it. Let G be a graph and let L = (L(v) : v ∈ V (G)) be a collection of lists. If each set L(v) is non-empty, then we say that L is a list assignment for G. If k is an integer and |L(v)| ≥ k for every v ∈ V (G), then we say that L is a k-list assignment for G. An. Thus if all the lists are the same, then this reduces to the notion of coloring. We say that a graph G is k-choosable, also called k-list-colorable, if G has an L-coloring for every k-list assignment L. The list chromatic...

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Section: Examples Pertaining To Exponentially Many Coloringsmentioning

confidence: 99%

Hyperbolic families and coloring graphs on surfaces

Postle

Thomas

2018

Trans. Amer. Math. Soc. Ser. B

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“…In particular, the arguments from Sections 3 imply that planar triangle‐free graphs have exponentially many colorings from lists of size four. Of course, there exist simpler proofs of this fact, see for example [11] for the triangle‐free case (even in a more general setting of graphs on surfaces).…”

Section: Flexibility and Reducible Configurationsmentioning

confidence: 99%

Flexibility of triangle‐free planar graphs

Dvořák

Masařík

Musílek

et al. 2020

Journal of Graph Theory

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“…Of course, there exist simpler proofs of this fact, see e.g. [10] for the triangle-free case (even in a more general setting of graphs on surfaces).…”

Section: Flexibility and Reducible Configurationsmentioning

confidence: 99%

Flexibility of triangle-free planar graphs

Dvořák,

Masařík,

Musílek

et al. 2019

Preprint

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Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle-free and all lists have size at least four, then there exists an L-coloring respecting at least a constant fraction of the preferences. * Work on this paper was supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.

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Exponentially many 4‐list‐colorings of triangle‐free graphs on surfaces

Cited by 4 publications

References 8 publications

Hyperbolic families and coloring graphs on surfaces

Hyperbolic families and coloring graphs on surfaces

Flexibility of triangle‐free planar graphs

Flexibility of triangle-free planar graphs

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