Related papers: Minimum-Turn Tours of Even Polyominoes

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of…

In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log…

We critically analyze the possibility of finding signatures of a phase transition by looking exclusively at static quantities of statistical systems, like e.g., the topology of potential energy sub-manifolds (PES). This topological…

Let $d\in {\mathbb N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb Z$, then $$\{(m,n)\in{\mathbb Z}^2: m+p_1(n),\ldots,m+p_d(n)\in S\}$$ is piecewise syndetic…

We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity…

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this…

In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical…

We show that a topologically minimal disk in a tetrahedron with index $n$ is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.

We discover some very general configuration results for constructing area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the…

A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered…

Haj\'os conjecture asserts that a simple Eulerian graph on n vertices can be decomposed into at most (n - 1)/2 cycles. The conjecture is only proved for graph classes in which every element contains vertices of degree 2 or 4. We develop new…

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in…

We present a Hamilton cycle in the $k$-sided pancake network and four combinatorial algorithms to traverse the cycle. The network's vertices are coloured permutations $\pi = p_1p_2\cdots p_n$, where each $p_i$ has an associated colour in…

For exact area-preserving twist maps, curves were constructed through the gaps of cantori in \cite{MMP84}, which were conjectured to have minimal flux subject to passing through the points of the cantorus. It was pointed out by \cite{Pol}…

In this study, a tight-binding model on square octagon lattice with nearest-neighbour and next-nearest-neighbour hoppings is considered. The system is topologically trivial although it exhibits quadratic band-touching points in its…

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell…

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product $G\leftthreetimes P,$ where $G$ is a compact…

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and…

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple…

A $triangulation$ is an embedding of a graph on surfaces where every face has length three. In this article, we show the existence of contractible Hamiltonian cycle in triangulated maps of which minimum degree is four.