Hyperbolic families and coloring graphs on surfaces

Abstract

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...

Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

“…Thomassen [27] naturally asked if such results could be generalized to list-coloring. For 5-list coloring, the author and Thomas [23] proved the analogous result while the author [19] proved the analogous result for 3-list-coloring graphs having girth at least five. These proofs are rather lengthy and rely on the theory of strongly hyperbolic families developed by the author and Thomas in [23].…”

“…For 5-list coloring, the author and Thomas [23] proved the analogous result while the author [19] proved the analogous result for 3-list-coloring graphs having girth at least five. These proofs are rather lengthy and rely on the theory of strongly hyperbolic families developed by the author and Thomas in [23]. Moreover, we proved that such critical graphs have at most O(g(Σ)) vertices, where g(Σ) denotes the Euler genus of Σ (that is 2h + c where h is the number of handles of Σ and c is the number of crosscaps).…”

“…We note that there are many graphs on surfaces which do have an L-coloring for every type 345 list assignment. In particular, the author and Thomas [23] showed that graphs with edge-width Ω(log g(Σ)) have this property.…”

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

“…The following theorem of Postle and Thomas utilizes this technique and extends Theorems and to graphs on surfaces. Theorem There exist constants such that the following holds. Let G be a graph on n vertices embedded in a fixed surface Σ of genus g , and let H be a proper subgraph of G .…”

“…Theorem 5. [4,5] There exist constants , > 0 such that the following holds. Let be a graph on vertices embedded in a fixed surface Σ of genus , and let be a proper subgraph of .…”