## Abstract

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 l 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∗(2^{n}). 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 Δ = 4,...,12 or with vector chromatic number at most 8. Our results extend the technique by Björklund [SIAM J. Comput., 43 (2014), pp. 280-299] and Björklund 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.

Original language | English |
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Pages (from-to) | 687-713 |

Number of pages | 27 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2017 |

Externally published | Yes |

## Keywords

- Algebraic techniques
- Coloring
- Fractional coloring
- Hamiltonian cycle
- K-Internal Spanning Tree
- K-Path
- Parameterized complexity
- Vector coloring

## ASJC Scopus subject areas

- General Mathematics