On consecutive edge magic total labeling of connected bipartite graphs

TL;DR

This paper investigates the properties of consecutive edge magic total labelings in connected bipartite graphs, identifying key constraints and extending existing results to specific graph classes like caterpillars and lobsters.

Contribution

It establishes that only four values of b allow for such labelings in connected bipartite graphs and extends previous findings to new graph subclasses.

Findings

Only four possible b-values for such labelings.

Extended results to caterpillars and lobsters.

Provided fundamental insights into bipartite graph labelings.

Abstract

Since Sedl\'{a}$\breve{\mbox{c}}$ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected bipartite graph. We make a very useful observation that there are only four possible values of $b$ for which a connected bipartite graph has a $b$-edge consecutive magic labeling. On the basis of this fundamental result, we deduce various interesting results on consecutive edge magic labelings of bipartite graphs, especially caterpillars and lobsters, which extends the results given by Sugeng and Miller.