Spanning trees with many leaves: new lower bounds in terms of number of vertices of degree~3 and at least~4

TL;DR

This paper establishes new lower bounds on the number of leaves in spanning trees of connected graphs based on the counts of vertices with degrees 3 and at least 4, and demonstrates the tightness of these bounds.

Contribution

It introduces a novel lower bound for leaves in spanning trees related to degree-3 and degree-4 vertices, including explicit exceptions and tightness proof.

Findings

Lower bound of (2/5)t + (1/5)s + α with α ≥ 8/5

α ≥ 2 for all but three regular degree-4 graphs

Constructed graph series show the bound's tightness

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.