Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and k-Path algorithms in undirected graphs by Björklund [FOCS’10], and Björklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O(1.6572) time. It can be applied as a subroutine to solve the k-Internal Spanning Tree (k-IST) problem in O(min(3.455, 1.946)) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of O(2). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree ∆ = 4, . . . , 12 or with vector chromatic number at most 8.