Improved bounds for spanning trees with many leaves

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n / 4 + 2 leaves and that this can be improved to ( n + 4 ) / 3 for cubic graphs without the diamond K 4 − e as a subgraph. We generalize the second result by proving that every graph G without diamonds and certain subgraphs called blossoms has a spanning tree with at least ( n ≥ 3 ( G ) + 4 ) / 3 leaves, where n ≥ 3 ( G ) is the number of vertices with degree at least 3 in G . We show that it is necessary to exclude blossoms in order to obtain a bound of the form n ≥ 3 ( G ) / 3 + c . This bound is used to deduce new similar bounds.