Degree Conditions for Completely Independent Spanning Trees of Bipartite Graphs

Let T_1,T_2,… , T_k be k(≥ 2) spanning trees of a graph G . The trees T_1,T_2,… , T_k are completely independent if the paths connecting any two vertices of G in these k trees are pairwise internally disjoint. Sufficient conditions for a graph to possess k completely independent spanning trees have attracted popular attention intensively. In this paper, we focus on such sufficient conditions for bipartite graphs, and show that a bipartite graph G=(X∪ Y, E) with order ν =m+n, m=|X|≥ 2k and n=|Y|≥ m, has k completely independent spanning trees if for every pair of vertices x∈ X,y∈ Y,d(x)≥(k-1)n/k+2,d(y)≥(k-1)m/k+2. Furthermore, we obtain that a bipartite graph G=(X∪ Y, E) with order ν≥ 8, |X|≥ 2k and |Y|≥ 2k, has k completely independent spanning trees if the minimum degree δ (G)≥⌊(3× 2^k-2-1)ν/3× 2^k-1⌋ +2, or δ (G)≥(k-1)ν/2k+2 and |X|=|Y|.