We give a systematic study of Hamiltonicity of grids-the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, "square", grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomial-time algorithm we give efficient algorithms to find a Hamiltonian cycle.We also establish, for any g ≥ 6, a one-to-one correspondence between Hamiltonian cycles in planar bipartite maximum-degree-3 graphs and Hamiltonian cycles in the class C g of girthg planar maximum-degree-3 graphs. As applications of the correspondence, we show that for graphs in C g the Hamiltonian cycle problem is NP-complete and that for any N ≥ 5 there exist graphs in C g that have exactly N Hamiltonian cycles. We also prove that for the graphs in C g , a Chinese Postman tour gives a (1 + 8 g )-approximation to TSP, improving thereby the Christofides ratio when g > 16. We show further that, on any graph, the tour obtained by Christofides' algorithm is not longer than a Chinese Postman tour.

Abstract.We consider contact representations of graphs where vertices are represented by cuboids, i.e. interior-disjoint axis-aligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a non-zero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axis-aligned 3D boxes. We prove that it is NP-complete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points,such that each M i is compatible with M i+1 . This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M 0 , M 1 ,. .. , M k = M , such that each M i is compatible with M i+1. This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.

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We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (ostinatos) from traditional world music. We prove that these Euclidean rhythms have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of evenness. We also show that essentially all Euclidean rhythms are deep: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval 1, 2, . . . , k − 1. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erdős in the plane.

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