Spanning -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 σ 2 ( G ) ≥ ( | G | − k + 2 ) / 2 , then G has a spanning k -ended tree, where σ 2 ( G ) denotes the minimum degree sum of two non-adjacent vertices of G . Moreover, the condition on σ 2 ( G ) is sharp. It was shown by Las Vergnas, and Broersma and Tuinstra, independently that if a graph H satisfies σ 2 ( 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.