Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008)

Exact coverings of 2-paths by hamilton cycles

AbstractWe construct a C(2m, 2m, 2) design which is a family of Hamilton cycles in K2m so that each 2-path of K2m lies in exactly two of the cycles

Black 1-factors and Dudeney sets

A set of Hamilton cycles in the complete graph Kn is called a Dudeney set if every path of length two lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set for every complete graph. It is known that there exists a Dudene

Uniform coverings of 2-paths with 4-cycles

Let G be a graph [a digraph] and H be a subgraph of G. A D(G,H,λ) design is a multiset D of subgraphs of G each isomorphic to H so that every 2-path [directed 2-path] of G lies in exactly λ subgraphs in D. In this paper, we show that there exists a D(Kn,n,C4,λ) design if and only if (i) n is even, or (ii) n is odd and λ is even. We also show that there exists a D(Kn,n∗,C⃗4,λ) design for every n and λ, where Kn,n and Kn,n∗ are the complete bipartite graph and the complete bipartite digraph, respectively; C4 and C⃗4 are a 4-cycle and a directed 4-cycle, respectively

3-perfect hamiltonian decomposition of the complete graph

Let n ≥ 5 be an odd integer and Kn the complete graph on n vertices. Let i be an integer with 2 ≤ i ≤ (n-1)/2. A hamiltonian decomposition H of Kn is called i-perfect if the set of the chords at distance i of the hamiltonian cycles in H is the edg

Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5]

Minimum Perimeter Developments of the Platonic Solids

A development of a convex polyhedron is a connected plane figure obtained by cutting the surface of the polyhedron and unfolding it. In this paper, we determine the length and configuration of a minimum perimeter development for each of the Platonic solids. We show that such developments are obtained by cutting the surface of the polyhedron along a Steiner minimal tree. We introduce the concept of Steiner isomorphism to develop a search algorithm for determining these Steiner minimal trees. Each of these trees is completely symmetric with respect to rotation around a fixed point