On the minimum leaf number of cubic graphs

Abstract

The minimum leaf numberml(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)≤n6+13, improving on the best known result in Salamon and Wiener (2008) and proving the conjecture in Zoeram and Yaqubi (2017). We further prove that if G is also 2-connected, then ml(G)≤n6.53, improving on the best known bound in Boyd et al. (2014). 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.