A Note On Vertex Distinguishing Edge colorings of Trees

A proper edge coloring of a simple graph G is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices u and v of G, the set of the colors assigned to the edges incident to u differs from the set of the colors assigned to the edges incident to v. The minimum number of colors required for all vdecs of G is denoted by χ ′ s (G) called the vdec chromatic number of G. Let nd(G) denote the number of vertices of degree d in G. In this note, we show that a tree T with n2(T ) ≤ n1(T ) holds χ ′ s (T ) = n1(T )+1 if its diameterD(T ) = 3 or one of two particular trees with D(T ) = 4, and χ ′ s(T ) = n1(T ) otherwise; furthermore χ ′ es(T ) = χ ′ s(T ) when |E(T )| ≤ 2(n1(T ) + 1), where χ ′ es (T ) is the equitable vdec chromatic number of T . AMS Subject Classification (2000): 05C15