Steinberg's Conjecture is false

Cohen-Addad, Vincent; Hebdige, Michael; Kráľ, Daniel; Li, Zhentao; Salgado, Esteban

doi:10.48550/arxiv.1604.05108

Cited by 2 publications

(3 citation statements)

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“…While Steinberg's conjecture was recently disproved by Cohen-Addad, Hebdige, Kral, Li and Salgado [15], Borodin, Glebov, Montassier and Raspaud [13] showed that every planar graph with no cycles of length four, five, six or seven is 3-colorable. That leads us to the following example.…”

Section: Examples Arising From (List-)coloringmentioning

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 Arising From (List-)coloringmentioning

confidence: 99%

Hyperbolic families and coloring graphs on surfaces

Postle

Thomas

2018

Trans. Amer. Math. Soc. Ser. B

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“…Steinberg [109] conjectured that every planar graph without 4-cycles or 5-cycles is 3colorable. Eventually, Cohen-Addad et al [51] found counterexamples. Results on this family can be compared with the family where mad(G) < 4; see Exercise 3.21.…”

Section: Discharging On Plane Graphsmentioning

confidence: 99%

“…During the 40 years between [109] and [51], many papers used discharging to prove 3colorability under various conditions excluding sets of cycle lengths. For example, Borodin et al [28] proved that excluding cycles of lengths 4 through 7 suffices.…”

Section: Discharging On Plane Graphsmentioning

confidence: 99%