ON THE LABEL GRAPHOIDAL COVERING NUMBER-II

Let G = (V, E) be a graph with p vertices and q edges. An acyclic graphoidal cover of G is a collection psi of paths in G which are internally disjoint and covering each edge of the graph exactly once. Let f : V -> {1, 2,..., p} be a labeling of the vertices of G. Let up arrow G(f) be the directed graph obtained by orienting the edges uv of G from u to v provided f(u) < f(v). If the set psi(f) of all maximal directed paths in up arrow G(f), with directions ignored, is an acyclic graphoidal cover of G, then f is called a graphoidal labeling of G and G is called a label graphoidal graph and eta(l) = min{vertical bar psi(f vertical bar) : f is a graphoidal labeling of G} is called the label graphoidal covering number of G. An orientation of G in which every vertex of degree greater than 2 is either a sink or a source is a graphoidal orientation. In this paper we characterize graphs for which (i) eta(l) = eta(a) and (ii) eta(l) = Delta. Also, we discuss the relation between graphoidal labeling and graphoidal orientation.