Spanning trees of bipartite graphs with a bounded number of leaves and branch vertices

Abstract

Let $T$ be a tree, a vertex of $T$ with degree one is often called a leaf of $T$, and a vertex of degree at least three is a branch vertex of $T$. We prove the following theorem. Let $k\ge 2$ be an integer and $G$ be a connected bipartite graph with bipartition $(A,B)$ such that $\left|A\right|\le \left|B\right|\le \left|A\right|+k-1$. If ${\sigma }_{1,1}\left(G\right)\ge \left|B\right|$, then $G$ has a spanning tree with few branch vertices and leaves, where ${\sigma }_{1,1}\left(G\right)$ denotes the minimum degree sum of two non-adjacent vertices taken from $A$ and $B$, respectively. Moreover, the condition on ${\sigma }_{1,1}\left(G\right)$ is sharp.