On the bipartite graph packing problem

The graph packing problem is a well-known area in graph theory. We consider a bipartite version and give almost tight conditions on the packability of two bipartite sequences

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 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, 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^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 LEAF SPANNING TREE.Comment: 25 pages, 27 Figure