Spanning kk-ended trees of bipartite graphs

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