Spanning trees with many leaves in graphs without diamonds and blossoms

Abstract

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 with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n∈+∈4)/3 leaves. 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 k ) 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-Leaves Spanning Tree. © 2008 Springer-Verlag Berlin Heidelberg.