Completely Independent Spanning Trees on Complete Graphs, Complete Bipartite Graphs and Complete Tripartite Graphs

Abstract

Let T ₁, T ₂,…, T _k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T ₁, T ₂,…, T _k are vertex-disjoint except end vertices x and y, then T ₁, T ₂,…, T _k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k-connected graph. Péterfalvi disproved the conjecture in 2012. In this paper, we shall prove that there are $\lfloor\frac{n}{2}\rfloor$ completely independent spanning trees in a complete graph with $n ( \geqslant 4)$ vertices. Then, we prove that there are $\lfloor\frac{n}{2}\rfloor$ completely independent spanning trees in a complete bipartite graph K _m,n where $m\geqslant n\geqslant 4$. Next, we also prove that there are $\lfloor\frac{n_1+n_2}{2}\rfloor$ completely independent spanning trees in a complete tripartite graph $K_{n_3,n_2,n_1}$ where $n_3\geqslant n_2\geqslant n_1$ and $n_1+n_2\geqslant 4$. As a result, the Hasunuma’s conjecture holds for complete graphs and complete m-partite graphs where m ∈ {2,3}.

This research was partially supported by National Science Council under the Grants NSC100-2628-E-141-001-MY2, NSC101-2221-E-131-039 and NSC101-2115-M-141-001.