Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1511.06568

Matchings of quadratic size extend to long cycles in hypercubes

Tomáš Dvořák
View original source ↗Read source document ↗Export metadata ↓

Abstract

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.

Explore related subjects

cs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tomáš Dvořák. 2016-08-29. Matchings of quadratic size extend to long cycles in hypercubes. https://doi.org/10.46298/dmtcs.1336

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Factorisability of Low Dimensional Non-Negative Integer Matrices

We consider the problem of determining if a given two-dimensional nonnegative integer matrix $M$ is the product of two such matrices, excluding trivial units. A matrix $M$ with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of $2 \times 2$ matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.

cs.DM↗
arXiv2026

The parameterised complexity of generalised temporal domination on temporal graphs with modular structure

Inspired by the static problem $(α,β)$-Dominating Set, we propose a general temporal domination problem, called $(α,β)$-Temporal Dominating Set ($(α,β)$-TDS). We show that this problem encompasses Temporal Dominating Set, and additionally provides first temporal extensions of problems such as $k$-Dominating Set and $α$-Dominating Set. In this paper, we study the parameterised complexity of $(α,β)$-TDS with respect to temporal neighbourhood diversity (TND), temporal modular-width (TMW), and temporal cliquewidth (TCW). We obtain fixed parameter tractability results for all values of $α$ and $β$ with respect to TND; W[1]-hardness with respect to TMW and TCW whenever $β$ is in the problem input, or whenever $α\in (0,1)$ and $β$ is a fixed constant; and para-NP-hardness with respect to TCW when $α= 0$ and $β= 1$, or $α= 1$ and $β= 0$.

cs.DM↗