Decomposition of hypercubes into sunlet graphs

TL;DR

This paper characterizes when hypercubes can be decomposed into sunlet graphs, providing necessary and sufficient conditions for certain dimensions and linking decompositions to cycle decompositions.

Contribution

It establishes the exact conditions for decomposing hypercubes into sunlet graphs and relates these decompositions to cycle decompositions in hypercubes.

Findings

Hypercube $Q_n$ admits an $L_{16}$-decomposition iff $n=4$ or $n extgreater=6$.

For any $m extgreater=2$, $Q_{mn}$ has an $L_{2k}$-decomposition if $Q_n$ has a $C_k$-decomposition.

Abstract

For any positive integer $k \geq 3,$ the sunlet graph of order $2k$, denoted by $L_{2k},$ is the graph obtained by adding a pendant edge to each vertex of a cycle of length $k.$ In this paper, we prove that the necessary and sufficient condition for the existence of an $L_{16}$-decomposition of the $n$-dimensional hypercube $Q_n$ is $n = 4$ or $n \geq 6.$ Also, we prove that for any integer $m \geq 2,$ $Q_{mn}$ has an $L_{2k}$-decomposition if $Q_{n}$ has a $C_k$-decomposition.