Abstract
An adjacent vertex-distinguishing edge coloring of a graph is a proper edge coloring such that no pair of adjacent vertices meets the same set of colors. The adjacent vertex-distinguishing edge chromatic number is the minimum number of colors required for an adjacent vertex-distinguishing edge coloring, denoted as \(\chi '_{as}(G)\). In this paper, we prove that for a connected graph G with maximum degree \(\Delta \ge 3\), \(\chi '_{as}(G)\le 3\Delta -1\), which proves the previous upper bound. We also prove that for a graph G with maximum degree \(\Delta \ge 458\) and minimum degree \(\delta \ge 8\sqrt{\Delta ln \Delta }\), \(\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }\).
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Acknowledgements
The authors are grateful to reviewers for useful remarks, and convey individual thanks to the referee for the thorough inspection of the proof. This work was supported by NSFC (Grant No.11771403).
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Zhu, J., Bu, Y. & Dai, Y. Upper bounds for adjacent vertex-distinguishing edge coloring. J Comb Optim 35, 454–462 (2018). https://doi.org/10.1007/s10878-017-0187-0
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DOI: https://doi.org/10.1007/s10878-017-0187-0