Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 454–462

# Upper bounds for adjacent vertex-distinguishing edge coloring

Article

## 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 $$\delta \ge 8\sqrt{\Delta ln \Delta }$$, $$\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }$$.

### Keywords

Proper edge coloring Adjacent vertex-distinguishing edge coloring Lov$$\acute{a}$$sz local lemma

## Notes

### 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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