Double-Star Decomposition of Regular Graphs

TL;DR

This paper extends the decomposition of regular graphs into double-stars from even to odd degrees, showing that certain odd-regular graphs with two disjoint perfect matchings can be decomposed into specific double-star subgraphs.

Contribution

It generalizes the edge-decomposition of regular graphs into double-stars from even to odd degrees, under the condition of containing two disjoint perfect matchings.

Findings

Every 2k-regular graph can be decomposed into double-stars.

Extension to (2k+1)-regular graphs with two disjoint perfect matchings.

Decomposition into S_{k_1, k_2} and S_{k_1-1, k_2} for all positive k_1, k_2 with k_1 + k_2= k.

Abstract

A tree containing exactly two non-pendant vertices is called a double-star. A double-star with degree sequence $(k_1+ 1, k_2+ 1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. We study the edge-decomposition of regular graphs into double-stars. It was proved that every double-star of size $k$ decomposes every $2k$-regular graph. In this paper, we extend this result to $(2k+ 1)$-regular graphs, by showing that every $(2k+ 1)$-regular graph containing two disjoint perfect matchings is decomposed into $S_{k_1, k_2}$ and $S_{k_{1}-1, k_2}$, for all positive integers $k_1$ and $k_2$ such that $k_1 + k_2= k$.