D-Lucky Labeling of Graphs

Abstract

Let l: V (G) →N be a labeling of the vertices of a graph G by positive integers. Define , where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u . In this paper we introduce a new labeling called d -lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d -lucky if c(u) ≠ c(v) , for every pair of adjacent vertices u and v in G . The d -lucky number of a graph G , denoted by η dl (G), is the least positive k such that G has a d -lucky labeling with {1,2, ..., k} as the set of labels. We obtain η dl (G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree.