Using path coloring of graphs for communication in social networks

Abstract

An L (p 1,  p 2,  p 3,   … ,  p m )- labeling of a graph G, has the vertices of G assigned with non-negative integers, such that the vertices at distance j should have at least p j as their label difference. If m = 3 and p 1 = 3,  p 2 = 2,  p 3 = 1, it is called an L (3,  2,  1)-labeling which is widely studied in the literature. In this paper, we define an L (3,  2,  1)-path coloring of G as a labeling g : V (G) → Z + such that between every pair of vertices there exists at least one path P where in the labeling restricted to this path is an L (3,  2,  1)-labeling. Among the labels assigned to any vertex of G under g, the maximum label is called the span of g. The L (3,  2,  1)-connection number of a graph G, denoted by k 3c  (G) is defined as the minimum value of span of g taken over all such labelings g. We call graphs with the special property that k 3c  (G) = |V (G) | as L (3,  2,  1)-path graceful. In this paper, we obtain k 3c  (G) of graphs that possess a Hamiltonian path and carry forward the discussion to certain classes of graphs which do not possess a Hamiltonian path, which is novel to this paper. Although different kinds of labeling are studied in the literature with different mathematical constraints imposed, the idea of showing the existence of a graph with a given number as its minimum labeling number has rarely been addressed. We show that given any positive integer, there always exists an L (3,  2,  1)-path graceful graph with the given integer as its k 3c  (G), thus addressing the inverse question. Finally exploiting the fact that there is no gap on the k 3c  (G) number line, we give an application of path colorings for secure communication on social networking sites. Efforts to deploy graph coloring in task scheduling, interference-free transmission, etc have been dealt by earlier researchers. In this paper, we deploy the L (3,  2,  1)-path coloring technique defined by us for secure communication in social networks, which has not been dealt with so far.