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arXiv · 1806.04451

On the minimum leaf number of cubic graphs

Jan Goedgebeur·Kenta Ozeki·Nico Van Cleemput·Gábor Wiener
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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.

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Jan Goedgebeur, Kenta Ozeki, Nico Van Cleemput, Gábor Wiener. 2018-06-12. On the minimum leaf number of cubic graphs. https://arxiv.org/abs/1806.04451

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