Spanning $k$-trees of Bipartite Graphs

A tree is called a k-tree if its maximum degree is at most k. We prove the following theorem. Let k ⩾ 2 be an integer, and G be a connected bipartite graph with bipartition (A,B) such that |A| ⩽ |B| ⩽ (k − 1)|A| + 1. If σk(G) ⩾ |B|, then G has a spanning k-tree, where σk(G) denotes the minimum degree sum of k independent vertices of G. Moreover, the condition on σk(G) is sharp. It was shown by Win (Abh. Math. Sem. Univ. Hamburg, 43, 63–267, 1975) that if a connected graph H satisfies σk(H) ⩾ |H|−1, then H has a spanning k-tree. Thus our theorem shows that the condition becomes much weaker if the graph is bipartite.