Neighbourly Regular Strength of Bipartite Graphs

A graph is said to be a neighbourly irregular graph (or simply an NI graph) if every pair of its adjacent vertices have distinct degrees. Let G be a simple graph of order n. Let NI(G) denote the set of all NI graphs in which G is an induced subgraph. The neighbourly regular strength of G is denoted by NRS(G) and is defined as the minimum positive integer k for which there is an NI graph in NI(G) of order n+k. In this paper, we show that NRS(G) ≤ 2 for any bipartite graph G. In addition, we show that NRS(T) is either 0 or 1 for any tree T.