The completely independent spanning trees in P4-free graphs

Hasunuma has conjectured that every $2k$-connected graph admits $k$ completely independent spanning trees. Pai et al. showed the conjecture is true for complete graphs, complete bipartite graphs and complete tripartite graphs, all of which are $P_4$-free. Chen et al. showed there exist two completely independent spanning trees in every 4-connected $P_4$-free graph. In this paper, we further discuss the completely independent spanning trees in both complete multipartite graphs and $P_4$-free graphs. Let $K_{n_1,n_2,\dots ,n_m}$ be a complete $m$-partite graph, where $n_1\le n_2\le \cdots \le n_m.$ We first prove that $K_{n_1,n_2,\dots ,n_m}$ contains $min\{\lfloor \frac{1}{2}\sum _{i=1}^{m-1}{n}_i\rfloor ,\lfloor \frac{1}{3}\sum _{i=1}^m{n}_i\rfloor \}$ completely independent spanning trees. This implies that every $2k$-connected complete $m$-partite graph admits $k$ completely independent spanning trees if its order is at least $3k.$ We further provide the upper and lower bounds of the maximum number of completely independent spanning trees in complete $m$-partite graphs, and demonstrate that these bounds are tight. Second, we show there exist $k$ completely independent spanning trees in every $2k$-connected $P_4$-free graph of order at least $3k.$ Finally, we construct a kind of minimum $P_4$-free graphs that contain $m$ completely independent spanning trees.