1,805 research outputs found

    Locally identifying coloring in bounded expansion classes of graphs

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    A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.]

    Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups

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    We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana

    Kernelization and Sparseness: the case of Dominating Set

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    We prove that for every positive integer rr and for every graph class G\mathcal G of bounded expansion, the rr-Dominating Set problem admits a linear kernel on graphs from G\mathcal G. Moreover, when G\mathcal G is only assumed to be nowhere dense, then we give an almost linear kernel on G\mathcal G for the classic Dominating Set problem, i.e., for the case r=1r=1. These results generalize a line of previous research on finding linear kernels for Dominating Set and rr-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on HH-topological-minor-free graphs. Also, we prove that for any somewhere dense class G\mathcal G, there is some rr for which rr-Dominating Set is W[22]-hard on G\mathcal G. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of rr-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded expansion graph

    On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs

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    For a positive integer kk, a proper kk-coloring of a graph GG is a mapping f:V(G)→{1,2,…,k}f: V(G) \rightarrow \{1,2, \ldots, k\} such that f(u)≠f(v)f(u) \neq f(v) for each edge uv∈E(G)uv \in E(G). The smallest integer kk for which there is a proper kk-coloring of GG is called chromatic number of GG, denoted by χ(G)\chi(G). A \emph{locally identifying coloring} (for short, lid-coloring) of a graph GG is a proper kk-coloring of GG such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer kk such that GG has a lid-coloring with kk colors is called \emph{locally identifying chromatic number} (for short, \emph{lid-chromatic number}) of GG, denoted by χlid(G)\chi_{lid}(G). In this paper, we study lid-coloring of Cartesian product and tensor product of two graphs. We prove that if GG and HH are two connected graphs having at least two vertices then (a) χlid(G□H)≤χ(G)χ(H)−1\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1 and (b) χlid(G×H)≤χ(G)χ(H)\chi_{lid}(G \times H) \leq \chi(G) \chi(H). Here G□HG \square H and G×HG \times H denote the Cartesian and tensor products of GG and HH respectively. We also give exact values of lid-chromatic number of Cartesian product (resp. tensor product) of two paths, a cycle and a path, and two cycles
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