735 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.]

    On the neighbour sum distinguishing index of planar graphs

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    Let cc be a proper edge colouring of a graph G=(V,E)G=(V,E) with integers 1,2,…,k1,2,\ldots,k. Then k≥Δ(G)k\geq \Delta(G), while by Vizing's theorem, no more than k=Δ(G)+1k=\Delta(G)+1 is necessary for constructing such cc. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger kk, i.e., k=Δ(G)+2k=\Delta(G)+2 enables enforcing additional strong feature of cc, namely that it attributes distinct sums of incident colours to adjacent vertices in GG if only this graph has no isolated edges and is not isomorphic to C5C_5. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph GG of maximum degree at least 2828 which contains no isolated edges admits a proper edge colouring c:E→{1,2,…,Δ(G)+1}c:E\to\{1,2,\ldots,\Delta(G)+1\} such that ∑e∋uc(e)≠∑e∋vc(e)\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e) for every edge uvuv of GG.Comment: 22 page
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