On Harmonious Labelings of the Balanced Quintuple Shells

Abstract

A multiple shell MS {n₁^t1, n₂^t2 ,. . . , n_r^tr } is a graph formed by t_i shells of widths n _i , 1 ≤ i ≤ r, which have a common apex. This graph has $\sum^{r}_{i=1}$ ti(ni–1)+1 vertices. A multiple shell is said to be balanced with width w if it is of the form MS{w^s} or MS{(w + 1)^t, w^s}. Deb and Limaye have conjectured that all multiple shells are harmonious. The conjecture has been shown true for the balanced double shells, the balanced triple shells and the balanced quadruple shells. In this paper, the conjecture is proved to be true for the balanced quintuple shells.