Discrete & Computational Geometry最新文献

Delone sets are discrete point sets X in ({mathbb {R}}^d) characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest “empty ball” that can be inserted into the interstices of X. The regularity radius ({hat{rho }}_d) is defined as the smallest positive number (rho ) such that each Delone set with congruent clusters of radius (rho ) is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that ({hat{rho }}_{d}={textrm{O}(d^2log _2 d)}R) as (drightarrow infty ), independent of r. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r and those with full-dimensional sets of d-reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that ({hat{rho }}_{d}={textrm{O}(dlog _2 d)}R) as (drightarrow infty ), independent of r.

The Sylvester–Gallai Theorem states that every rank-3 real-representable matroid has a two-point line. We prove that, for each (kge 2), every complex-representable matroid with rank at least (4^{k-1}) has a rank-k flat with exactly k points. For (k=2), this is a well-known result due to Kelly, which we use in our proof. A similar result was proved earlier by Barak, Dvir, Wigderson, and Yehudayoff and later refined by Dvir, Saraf, and Wigderson, but we get slightly better bounds with a more elementary proof.

It is broadly known that any parallelepiped tiles space by translating copies of itself along its edges. In earlier work relating to higher-dimensional sandpile groups, the second author discovered a novel construction which fragments the parallelepiped into a collection of smaller tiles. These tiles fill space with the same symmetry as the larger parallelepiped. Their volumes are equal to the components of the multi-row Laplace determinant expansion, so this construction only works when all of these signs are non-negative (or non-positive). In this work, we extend the construction to work for all parallelepipeds, without requiring the non-negative condition. This naturally gives tiles with negative volume, which we understand to mean canceling out tiles with positive volume. In fact, with this cancellation, we prove that every point in space is contained in exactly one more tile with positive volume than tile with negative volume. This is a natural definition for a signed tiling. Our main technique is to show that the net number of signed tiles doesn’t change as a point moves through space. This is a relatively indirect proof method, and the underlying structure of these tilings remains mysterious.

We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained for a fixed number of crossings. A random meander diagram is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. We use this for exploring geometric properties of a link complement. Specifically, we give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger’s algorithm.

Given a graph (G=(V,E)) and a set (S subseteq left( {begin{array}{c}V 2end{array}}right) ) of terminal pairs, the minimum multicut problem asks for a minimum edge set (delta subseteq E) such that there is no s-t-path in (G -delta ) for any ({s,t}in S). For (|S|=1) this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope (textsc {MultC}^square (G,S)) is the convex hull of all multicuts in G; the multicut dominant is given by (textsc {MultC}(G,S)=textsc {MultC}^square (G,S)+mathbb {R}^E_{{ge 0}}). The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.

We introduce the multi-width of a lattice polytope and use this to classify and count all lattice tetrahedra with multi-width ((1,w_2,w_3)). The approach used in this classification can be extended into a computer algorithm to classify lattice tetrahedra of any given multi-width. We use this to classify tetrahedra with multi-width ((2,w_2,w_3)) for small (w_2) and (w_3) and make conjectures about the function counting lattice tetrahedra of any multi-width.

A set ({mathcal {G}}) of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph (G in {mathcal {G}}) admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most ((3+o(1))log _2(n)) planar graphs on n vertices. This constitutes a double-exponential improvement over the previously best known bound of (O(ncdot 4^{n/11})) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer (nin [107,193]) there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size (O(log n)) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than (n^6) planar n-vertex graphs with no simultaneous embedding.

Normal complexes are orthogonal truncations of simplicial fans. In this paper, we develop the study of mixed volumes for normal complexes. Our main result is a sufficiency condition that ensures when the mixed volumes of normal complexes associated to a given fan satisfy the Alexandrov–Fenchel inequalities. By specializing to Bergman fans of matroids, we give a new proof of the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov–Fenchel inequalities for normal complexes.

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about (tfrac{4}{pi }). We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.