132 research outputs found

    Minimal counterexamples and discharging method

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    Recently, the author found that there is a common mistake in some papers by using minimal counterexample and discharging method. We first discuss how the mistake is generated, and give a method to fix the mistake. As an illustration, we consider total coloring of planar or toroidal graphs, and show that: if GG is a planar or toroidal graph with maximum degree at most κ−1\kappa - 1, where κ≥11\kappa \geq 11, then the total chromatic number is at most κ\kappa.Comment: 8 pages. Preliminary version, comments are welcom

    Counting and Enumerating Crossing-free Geometric Graphs

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    We describe a framework for counting and enumerating various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time O(2nn2)O(2^nn^2) where nn is the number of points. The main idea is to reduce the problem of counting geometric graphs to counting source-sink paths in a directed acyclic graph. The following new results will emerge. The number of all crossing-free geometric graphs can be computed in time O(cnn4)O(c^nn^4) for some c<2.83929c < 2.83929. The number of crossing-free convex partitions can be computed in time O(2nn4)O(2^nn^4). The number of crossing-free perfect matchings can be computed in time O(2nn4)O(2^nn^4). The number of convex subdivisions can be computed in time O(2nn4)O(2^nn^4). The number of crossing-free spanning trees can be computed in time O(cnn4)O(c^nn^4) for some c<7.04313c < 7.04313. The number of crossing-free spanning cycles can be computed in time O(cnn4)O(c^nn^4) for some c<5.61804c < 5.61804. With the same bounds on the running time we can construct data structures which allow fast enumeration of the respective classes. For example, after O(2nn4)O(2^nn^4) time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in nn. All described algorithms are comparatively simple, both in terms of their analysis and implementation

    Master index to volumes 251-260

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    Total coloring of planar graphs without some chordal 6-cycles

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    A k-total-coloring of a graph G is a coloring of vertex set and edge set using k colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if G is a planar graph with maximum ∆ ≥ 8 and every 6-cycle of G contains at most one chord or any chordal 6-cycles are not adjacent, then G has a (∆ + 1)-total-coloring

    A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm

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    We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.Comment: 37 pages (LaTeX2e). Includes tex file and 3 additional style files. The tex file includes 14 figures using pstricks.sty. Minor changes. Version published in J. Phys.
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