On Double-Star Decomposition of Graphs

A tree containing exactly two non-pendant vertices is called a double-star. A double-star with degree sequence $(k_1 + 1, k_2 + 1, 1, . . ., 1)$ is denoted by $ S_{k_1,k_2} $. We study the edge-decomposition of 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 by showing that every graph in which every vertex has degree $ 2k + 1 $ or $ 2k + 2 $ and containing a 2-factor 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 $k_1 + k_2 = k$.