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.