We show two results related to the Hamiltonicity and k-PATH algorithms in undirected graphs by Bjoerklund [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~*(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 Δ = 4,.... 12 or with vector chromatic number at most 8.
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