Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

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 K4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3 + c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m) + O(6.75) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for Max Leaf Spanning Tree.