Double Shells with Two Pendant Edges at the Apex are $k$-graceful

Abstract

A double shell is defined to be two edge-disjoint shells with a common apex. In this paper, we prove that double shells (where the shell orders are $m$ and $2m+1$) with exactly two pendant edges at the apex are $k$-graceful when $k=2$. We extend this result to double shells of any order $m$ and $\ell$ (where $m\ge 3$ and $\ell \ge 3$) with exactly two pendant edges at the apex.