A comprehensive survey on prime graphs

Abstract

This article presents a short and concise survey on prime graphs. The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications like “coding theory, x - ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management”. An assignment of integers to the vertices or edges or to both of a graph G(V,E) subject to certain constraints is called a graph labeling. The notion of “prime labeling” was originated by Entringer and considered in a paper by Tout, Dabboucy, and Howalla. A bijection f:V(G) → {1,2,3,…,|V|} is called a prime labeling of G if for each edge e = st, GCD(f(s),f(t)) = 1. where GCD denotes the greatest common divisor. We call G a prime graph if it admits a prime labeling. This article stands divided into six sections. The first and sixth sections are reserved respectively for introduction and a few important references. Sections 2, 3, and 4, respectively deal with the prime labeling of certain classes of graphs such as path, cycle, complete graph, complete bipartite graph, bipartite graphs, join and product graphs, wheel related graphs etc. wherein some known results of high importance have been recalled. The fifth section deals with the enumeration of conjectures and open problems in respect of prime labeling that still remain unsolved.