Spanning k-ended trees of bipartite graphs.

A tree is called a k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. We prove the following theorem. Let k≥2 be an integer, and let G be a connected bipartite graph with bipartition (A,B) such that |A|≤|B|≤|A|+k−1. If σ2(G)≥(|G|−k+2)/2, then G has a spanning k-ended tree, where σ2(G) denotes the minimum degree sum of two non-adjacent vertices of G. Moreover, the condition on σ2(G) is sharp. It was shown by Las Vergnas, and Broersma and Tuinstra, independently that if a graph H satisfies σ2(H)≥|H|−k+1 then H has a spanning k-ended tree. Thus our theorem shows that the condition becomes much weaker if a graph is bipartite.