SPOTTING TREES WITH FEW LEAVES

We show two results related to finding trees and paths in graphs. First, we show that in O* (1.657(k)2(l/2)) time one can either find a k-vertex tree with 1 leaves in an n-vertex undirected graph or conclude that such a tree does not exist. Our solution can be applied as a subroutine to solve the k-INTERNAL SPANNING TREE 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*(2n). Second, we show that the running time 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 HAMILTONICITY and k-PATH in any graph of maximum degree Delta = 4,..., 12 or with vector chromatic number at most 8. Our results extend the technique by Bjorklund [SIAM J. Comput., 43 (2014), pp. 280-299] and Bjorklund et al. [Narrow Sieves for Parameterized Paths and Packings, CoRR, arXiv:1007. 1161, 2010] to finding structures more general than paths as well as refine it to handle special classes of graphs more efficiently.