Bounds of a number of leaves of spanning trees

TL;DR

This paper establishes new lower bounds on the number of leaves in spanning trees of connected graphs based on vertex degrees, girth, and chain length, with proofs and exact bounds demonstrated through examples.

Contribution

It introduces novel bounds for leaves in spanning trees considering degree constraints, girth, and chain length, with proofs and tight examples.

Findings

Lower bound for leaves based on vertices not degree 2

Bounds depend on girth and chain length parameters

Examples show bounds are tight and exact

Abstract

We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge 1$. We prove that $G$ has a spanning tree with at least $\alpha_{g,k}(v(G)-k-2)+2$ leaves, where $\alpha_{g,k}= {[{g+1\over2}]\over [{g+1\over2}](k+3)+1}$ for $k<g-2$; $\alpha_{g,k}= {g-2\over (g-1)(k+2)}$ for $k\ge g-2$. We present infinite series of examples showing that all these bounds are exact.