On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V,E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u) ≠ C(v) for ∨uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2, … k, we have ‖E _i | ? |E _j ‖ ≤ 1, where E _i = {e|e ∈ E(G) and f(e) = i}. χ′ _ave (G) =min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χ′ _ave (G) of some special graphs and present a conjecture.