Towards Obtaining a 3-Decomposition From a Perfect Matching

A decomposition of a graph is set subgraphs whose edges partition those $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic can be decomposed into spanning tree, 2-regular subgraph, and matching. It has been settled for special classes graphs, one the first results being Hamiltonian graphs. In past two years several new have obtained, adding plane, claw-free, 3-connected tree-width 3 graphs to list. this paper, we regard natural extension graphs: removing cycle from leaves perfect Conversely, matching $M$ $G$ disjoint union cycles. Contracting these cycles yields $G_M$. star-like if $G_M$ star some $M$, making star-like. We extend technique used prove satisfy show it as well.