Decomposition of Augmented Cubes into Regular Connected Pancyclic Subgraphs

TL;DR

This paper demonstrates that augmented cubes can be decomposed into two spanning, regular, connected, and pancyclic subgraphs with specific degree and connectivity properties, enhancing understanding of their structural robustness.

Contribution

It provides a new decomposition method for augmented cubes into regular, connected, and pancyclic subgraphs, extending previous knowledge on their structural properties.

Findings

Augmented cubes can be decomposed into two regular, connected subgraphs for n ≥ 4.

Each subgraph can be made pancyclic if the degree is at least 3.

The decomposition maintains spanning and connectivity properties.

Abstract

In this paper, we consider the problem of decomposing the augmented cube $AQ_n$ into two spanning, regular, connected and pancyclic subgraphs. We prove that for $ n \geq 4$ and $ 2n - 1 = n_1 + n_2 $ with $ n_1, n_2 \geq 2,$ the augmented cube $ AQ_n$ can be decomposed into two spanning subgraphs $ H_1$ and $ H_2$ such that each $ H_i$ is $n_i$-regular and $n_i$-connected. Moreover, $H_i$ is $4$-pancyclic if $ n_i \geq 3.$