Spotting Trees with Few Leaves

Abstract

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 $O^{\ast }(1.657^k{2}^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^{\ast }(\min (3.455^k,1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^{\ast }(2^n)$. 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 $\Delta =4,\dots ,12$ or with vector chromatic number at most 8.

References in corpus (1)

• An upper bound on the fractional chromatic number of triangle-free subcubic graphs