On the minimum leaf number of cubic graphs

TL;DR

This paper improves bounds on the minimum leaf number of cubic graphs, proving conjectures and establishing new results for connected and 2-connected cases, with implications for graph spanning trees.

Contribution

It presents improved bounds on the minimum leaf number of cubic graphs, confirming conjectures and introducing new conjectures with tight bounds.

Findings

For connected cubic graphs, ml(G) ≤ n/6 + 1/3.

For 2-connected cubic graphs, ml(G) ≤ n/6.53.

New conjectures and tight examples for various cubic graph classes.

Abstract

The \emph{minimum leaf number} $\hbox{ml} (G)$ of a connected graph $G$ is defined as the minimum number of leaves of the spanning trees of $G$. We present new results concerning the minimum leaf number of cubic graphs: we show that if $G$ is a connected cubic graph of order $n$, then $\mathrm{ml}(G) \leq \frac{n}6 + \frac13$, improving on the best known result in [Inf. Process. Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph Theory and Applications 5 (2017) 207-211]. We further prove that if $G$ is also 2-connected, then $\mathrm{ml}(G) \leq \frac{n}{6.53}$, improving on the best known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new conjectures concerning the minimum leaf number of several types of cubic graphs and examples showing that the bounds of the conjectures are best possible.