Psykisk helsearbeid

Myrvold, Trine; Helgesen, Marit

doi:10.7577/nibr/notat/2005/108

Cited by 1 publication

(4 citation statements)

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“…This is an obvious induction on γ. For γ = 2 the statement is true by Myrvold's theorem [17]. Note that the deletion of an edge in a graph reduces the genus by at most one, which implies the rest.…”

Section: On the Parameter Mmentioning

confidence: 90%

“…S = N γ . To our knowledge, M(S) has been determined for the surfaces S of nonnegative Euler characteristics and the double torus only: M(S 1 ) = M(N 1 ) = 9 by Kuratowski's Theorem, M(N 2 ) = 15 by results of Glover, Huneke and Wang [8] and Archdeacon [2], and M(S 2 ) = 18 by work of Myrvold [17]. Note that it is not even known whether M(S γ+1 ) = M(N 2γ+1 ).…”

Section: The Structure Of Digraphs In Surfacesmentioning

confidence: 99%

“…Myrvold [17] gives a classification of all irreducible graphs for the torus with at most 11 vertices. These graphs have at least 18 edges.…”

Section: On the Parameter Mmentioning

confidence: 99%

“…For the torus and the projective plane this parameter is 9 by the Kuratowski theorem. It is 15 for the Klein bottle by [8,2] and 18 for the double torus by [17]. In general, the better the topological parameter can be estimated, the better will be the bounds for lightness, and the more surfaces our results will apply to.…”

Section: Introductionmentioning

confidence: 98%

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Lightness of digraphs in surfaces and directed game chromatic number

Andres

2009

Discrete Mathematics

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The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k, and without 4-cycles, which can be embedded in a surface S. (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k ≥ 5, these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [11] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game coloring number, and thus for the game chromatic number of (di)graphs with girth k and without 4-cycles embeddable in S. The game chromatic resp. game coloring number were introduced by Bodlaender [3] resp. Zhu [22] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game coloring number of a directed simple forest is at most 3.

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Section: On the Parameter Mmentioning

confidence: 90%