148 research outputs found

    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

    Locally identifying colourings for graphs with given maximum degree

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    A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that any graph G has a locally identifying colouring with 2Δ2−3Δ+32\Delta^2-3\Delta+3 colours, where Δ\Delta is the maximum degree of G, answering in a positive way a question asked by Esperet et al. We also provide similar results for locally identifying colourings which have the property that the colours in the neighbourhood of each vertex are all different and apply our method to the class of chordal graphs
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