Decomposition of wheel graphs into stars, cycles and paths

Abstract

Let $G=(V, E)$ be a finite graph with $n$ vertices. The Wheel graph $W_n$ is a graph with vertex set $\left\{v_0, v_1, v_2, \ldots, v_n\right\}$ and edge-set consisting of all edges of the form $v_i v_{i+1}$ and $v_0 v_i$ where $1 \leq i \leq n$, the subscripts being reduced modulo $n$. Wheel graph of $(n+1)$ vertices denoted by $W_n$. Decomposition of Wheel graph denoted by $D\left(W_n\right)$. A star with 3 edges is called a claw $S_3$. In this paper, we show that any Wheel graph can be decomposed into following ways.
$$
D\left(W_n\right)=\left\{\begin{array}{ll}
(n-2 d) S_3, d=1,2,3, \ldots & \text { if } n \equiv 0(\bmod 6) \\
{[(n-2 d)-1] S_3 \text { and } P_3, d=1,2,3 \ldots} & \text { if } n \equiv 1(\bmod 6) \\
{[(n-2 d)-1] S_3 \text { and } P_2, d=1,2,3, \ldots} & \text { if } n \equiv 2(\bmod 6) \\
(n-2 d) S_3 \text { and } C_3, d=1,2,3, \ldots & \text { if } n \equiv 3(\bmod 6) \\
(n-2 d) S_3 \text { and } P_3, d=1,2,3 \ldots & \text { if } n \equiv 4(\bmod 6) \\
(n-2 d) S_3 \text { and } P_2, d=1,2,3, \ldots & \text { if } n \equiv 5(\bmod 6)
\end{array} .\right.
$$