11,690 research outputs found

    Edge-coloring of split graphs

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    Orientador: Célia Picinin de MelloTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Por apresentar basicamente fórmulas, o resumo, na íntegra, poderá ser visualizado no texto completo da tese digitalAbstract: Not informedDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

    Split-critical and uniquely split-colorable graphs

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    The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic

    New Results on Edge-coloring and Total-coloring of Split Graphs

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    A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph GG is said to be tt-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most tt. Given a graph GG, determining the smallest tt for which GG is tt-admissible, i.e. the stretch index of GG denoted by σ(G)\sigma(G), is the goal of the tt-admissibility problem. Split graphs are 33-admissible and can be partitioned into three subclasses: split graphs with σ=1,2\sigma=1, 2 or 33. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with Δ\Delta or Δ+1\Delta+1 colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, i.e., a total coloring of GG, it is conjectured that any graph can be total colored with Δ+1\Delta+1 or Δ+2\Delta+2 colors, and thus can be classified as Type 1 or Type 2. These both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with σ=2\sigma=2. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.Comment: 20 pages, 5 figure

    Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

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    A path in an edge-colored graph GG is rainbow if no two edges of it are colored the same. The graph GG is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph GG is strongly rainbow-connected. The minimum number of colors needed to make GG rainbow-connected is known as the rainbow connection number of GG, and is denoted by rc(G)\text{rc}(G). Similarly, the minimum number of colors needed to make GG strongly rainbow-connected is known as the strong rainbow connection number of GG, and is denoted by src(G)\text{src}(G). We prove that for every k3k \geq 3, deciding whether src(G)k\text{src}(G) \leq k is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an nn-vertex split graph with a factor of n1/2ϵn^{1/2-\epsilon} for any ϵ>0\epsilon > 0 unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure
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