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}.Keywordscompletely independent spanning treesedge-disjoint spanning treescomplete graphscomplete bipartite graphscomplete tripartite graphs