Upper bounds for adjacent vertex-distinguishing edge coloring

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 $$\chi '_{as}(G)$$ χ a s ′ ( G ) . In this paper, we prove that for a connected graph G with maximum degree $$\Delta \ge 3$$ Δ ≥ 3 , $$\chi '_{as}(G)\le 3\Delta -1$$ χ a s ′ ( G ) ≤ 3 Δ - 1 , which proves the previous upper bound. We also prove that for a graph G with maximum degree $$\Delta \ge 458$$ Δ ≥ 458 and minimum degree $$\delta \ge 8\sqrt{\Delta ln \Delta }$$ δ ≥ 8 Δ l n Δ , $$\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }$$ χ a s ′ ( G ) ≤ Δ + 1 + 5 Δ l n Δ .