Bing YAO, Zhong-fu ZHANG, Jian-fang WANG
A k-edge-coloring f of a connected graph G is a (λ1, λ2, …,λβ)-defected k-edge-coloring if there is a smallest integer β with 1≤β≤k-1 such that the multiplicity of each color j∈{1, 2, …, β} appearing at a vertex is equal to λj≥2, and each color of {β + 1, β+ 2.…, k} appears at some vertices at most one time. The (λ1, λ2, …,λβ)-defected chromatic index of G, denoted as x'(λ1, λ2, …,λβ;G), is the smallest number such that every (λ1, λ2, …,λβ)-defected t-edge-coloring of G holds t≥ x'(λ1, λ2, …,λβ;G). We obtain Δ(G) ≤ x'(λ1, λ2, …,λβ;G) + ？？？777？？？(λi-1)≤ Δ(G) + 1, and introduce two new chromatic indices of G as: the vertex pan-biuniform chromatic index xpb'(G), and the neighbour vertex pan-biuniform chromatic index xnpb'(G), and furthermore find the structure of a tree T having xpb'(T) = 1.
