On Equitable Vertex Distinguishing Edge Colorings of Trees

Abstract

It has been known that determining the exact value of vertex distinguishing edge index χs′(G) of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let nd(G) denote the number of vertices of degree d in G, and let χ′es(G) be the equitable vertex distinguishing edge index of G. We show that a tree T holds n1(T) ≤ χs′(T) ≤ n1(T) + 1 and χs′(T) = χes′(T) if T satisfies one of the following conditions (i) n2(T) ≤ Δ(T) or (ii) there exists a constant c with respect to 0 <c < 1 such that n2(T) ≤cn1(T) and ∑3≤d≤Δ(T)nd(T) ≤ (1−c)n1(T)+1.