γ-Max Labelings of Graphs with Exterior Major Vertices

Let G be a graph of order n and size in. A gamma-labeling of G is a one-to-one function f : V (G) -> {0,1,2, ... , m} that induces an edge-labeling f' : E(G) -> {1, 2, , m} on G defined by & para;& para;f' (e) = vertical bar f (u) - f (v)vertical bar, for each edge e = uv in E(G) .& para;& para;The value of f is defined as val(f) = Sigma(e is an element of E(G)) f' (e). The maximum value of a gamma-labeling of G is defined as & para;& para;val(max) (G) = max{val(f) vertical bar f is a gamma-labeling of G}.& para;& para;For a gamma-labeling f of a graph G, a gamma-orientation D(f) of f is an oriented graph derived from a gamma-labeling f of G, by assigning to each edge xy the orientation (x, y) if f(x) < f(y).& para;& para;A vertex of degree at least 3 in a graph G is called a major vertex. The major degree ma(G) of a graph G is the number of major vertices of G. An end-vertex z of G is said to he a terminal vertex of a major vertex v of G if d(z, v) < d(z, w) for every other major vertex w of G. A major vertex v of a graph G is an exterior major vertex of G if it has at least one terminal vertex.& para;& para;In this paper, we characterize a gamma-orientation D(f) of a gamma-max labeling f of a graph G with exterior major vertices. Furthermore, we determine the maximum value of a gamma-labeling of a tree T with a unique exterior major vertex and also a tree T of ma(T) = 2 with adjacent exterior major vertices.