An Algorithm for the Decomposition of Complete Graph into Minimum Number of Edge-disjoint Trees

In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=\lceil \frac{n}{2}\rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M=\{m_1,m_2,\mathrm{.}\mathrm{.}\mathrm{.},m_x\}$ where $m_i\le (n-1)$ and ${\mathrm{\Sigma }}_{i=1}^x{m}_i$ = $\frac{n(n-1)}{2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.