Spanning trees with many leaves: lower bounds in terms of number of vertices of degree 1, 3 and at least~4

TL;DR

This paper establishes a lower bound on the number of leaves in spanning trees of connected graphs based on the counts of vertices with degrees 1, 3, and at least 4, and demonstrates the tightness of this bound.

Contribution

It provides a new lower bound for leaves in spanning trees related to specific degree distributions and proves its tightness with infinite graph series.

Findings

Lower bound of (1/3)t + (1/4)s + 3/2 leaves in spanning trees

Bound is tight, shown by infinite series of graphs

Applicable to connected graphs with specified degree vertices

Abstract

We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that our bound is tight.