On Harmonious Labelings of the Balanced Quadruple Shells

Abstract

A multiple shell \(MS\{n_1^{t_1}, n_2^{t_2}, \dots, n_r^{t_r}\}\) is a graph formed by \(t_i\) shells of widths \(n_i\), \(1 \leq i \leq r\), which have a common apex. This graph has \(\sum_{i=1}^rt_i(n_i-1) + 1\) vertices. A multiple shell is said to be balanced with width \(w\) if it is of the form \(MS\{w^t\}\) or \(MS\{w^t, (w+1)^s\}\). Deb and Limaye have conjectured that all multiple shells are harmonious, and shown that the conjecture is true for the balanced double shells and balanced triple shells. In this paper, the conjecture is proved to be true for the balanced quadruple shells.