Estimating the circumference of a graph in terms of its leaf number

TL;DR

This paper investigates the relationship between the leaf number of a graph and its circumference, establishing bounds and conditions under which the graph is Hamiltonian, especially for regular graphs.

Contribution

It introduces bounds on the circumference based on leaf number and minimum degree, and proves Hamiltonicity for regular graphs under these conditions.

Findings

Circumference of G is at least n-1.

If G is regular, then G is Hamiltonian.

L(G) ≤ 2δ - 1 implies certain cycle properties.

Abstract

Let $\mathcal{T}$ be the set of spanning trees of $G$ and let $L(T)$ be the number of leaves in a tree $T$. The leaf number $L(G)$ of $G$ is defined as $L(G)=\max\{L(T)|T\in \mathcal{T}\}$. Let $G$ be a connected graph of order $n$ and minimum degree $\delta$ such that $L(G)\leq 2\delta-1$. We show that the circumference of $G$ is at least $n-1$, and that if $G$ is regular then $G$ is hamiltonian.