Pack Graphs with Subgraphs of Size Three

Abstract

An $H$-packing $\mathcal{F}$ of a graph $G$ is a set of edge-disjoint subgraphs of $G$ in which each subgraph is isomorphic to $H$. The leave $L$ or the remainder graph $L$ of a packing $\mathcal{F}$ is the subgraph induced by the set of edges of $G$ that does not occur in any subgraph of the packing $\mathcal{F}$. If a leave $L$ contains no edges, or simply $L = \phi$, then $G$ is said to be $H$-decomposable, denoted by $H \mid G$. In this paper, we prove a conjecture made by Chartrand, Saba and Mynhardt [13]: If $G$ is a graph of size $q(G) \equiv 0 \pmod{3}$ and $\delta(G) \geq 2$, then $G$ is $H$-decomposable for some graph $H$ of size $3$.