Spotting Trees with Few Leaves

Abstract

We show two results related to the Hamiltonicity and \(k\) -Path algorithms in undirected graphs by Bjorklund [FOCS’10], and Bjorklund 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 \(O^*(1.657^k2^{l/2})\) time. It can be applied as a subroutine to solve the \(k\) -Internal Spanning Tree (\(k\)-IST) problem in \(O^*({\text {min}}(3.455^k, 1.946^n))\) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time, we break the natural barrier of \(O^*(2^n)\). 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 \(\Delta =4,\ldots ,12\) or with vector chromatic number at most \(8\).