On the minimum leaf number of cubic graphs
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The minimum leaf number 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 ml(G) ≤n/6 + 1/3, 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 ml(G) ≤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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