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    Progress on the adjacent vertex distinguishing edge colouring conjecture

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    A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree Δ\Delta and no isolated edge has an adjacent vertex distinguishing edge colouring with Δ+300\Delta + 300 colours, provided Δ\Delta is large enough. We show that this bound can be reduced to Δ+19\Delta + 19. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that Δ+2\Delta + 2 colours are enough for Δ≥3\Delta \geq 3.Comment: v2: Revised following referees' comment
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