Spanning Trees with many Leaves: New Lower Bounds in Terms of the Number of Vertices of Degree 3 and at Least 4

Abstract

We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for all graphs besides three exclusions. All exclusion are regular graphs of degree~4, they are explicitly described in the paper. We present infinite series of graphs, containing only vertices of degrees~3 and~4, for which the maximal number of leaves in a spanning tree is equal for ${2\over 5}t +{1\over 5}s+2$. Therefore we prove that our bound is tight.