Pavel Valtr
We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.
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We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].
An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.
In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.
Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\lfloor\frac{n}{10}\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, tw…
Rotationally Symmetric Tilings with Convex Pentagons and Hexagons
本文研究具有旋转对称性的平面铺砌，引入一类特定的凸五边形，证明对于任意对称类型 $\mathbf{C}_{n}$ 或 $\mathbf{D}_{n}$ 均存在由该类五边形生成的单面铺砌，且 $n>1$ 时为 $n$ 臂螺旋铺砌，该结论同样适用于凸六边形。
Happy Ending: An Empty Hexagon in Every Set of 30 Points
该研究借助可满足性求解与紧凑编码、搜索空间划分策略，证明平面中任意30个一般位置点必含空六边形，确立精确界限。
For each $d>0$, we find all the smallest fullerenes for which the least distance between two pentagons is $d$. We also show that for each $d$ there is an $h_d$ such that fullerenes with pentagons at least distance $d$ apart and any number of hexagons greater than or equal to $h_d$ exist. We also determine the number of fullerenes where the minimum distance between any two pentagons is at least $d$, for $1 \le d \le 5$, up to 400 vertices.
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.
In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.
Polyhedra with hexagonal and triangular faces and three faces around each vertex
本文研究了由六边形和三角形组成且每个顶点有三个面的3-正则平面图（trihexes），通过signature标记法建立了其与等价类的双射，并据此限制了给定顶点数下的trihexes数量及证明了无六边形“belts”的猜想。
An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is non-empty and unique. Previously, $r$-identifying codes have been studied in various grids. In particular, it has been shown that there exists a 2-identifying code in the hexagonal grid with density 4/19 and that there are no 2-identifying codes with density smaller than 2/11. Recently, the lower bound has been improved to 1/5 by Martin and Stanton (2010). In this paper, we prove that the 2-identifying code with density 4/…
We prove that a non-empty set L of at most q^5+q^4+q^3+q^2+q+1 lines of PG(n, q) with the properties that (1) every point of PG(n,q) is incident with either 0 or q+1 elements of L, (2) every plane plane of PG(n, q) is incident with either 0, 1 or q+1 elements of L, (3) every solid of PG(n, q) is incident with either 0, 1, q+1 or 2q+1 elements of L, and (4) every 4-dimensional subspace of PG(n, q) is incident with at most q^3-q^2+4q elements of L, is necessarily the set of lines of a split Cayley hexagon H(q) naturally embedded in PG(6, q).
For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets $N[v_1] \cap D$ and $N[v_2] \cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \approx 0.413793$ and was pr…
There are many 5-holes
研究证明平面中任意n个一般位置点集至少包含Ω(n^{20/11})个空凸五边形，改进了此前结果并缩小了与猜想Ω(n²)下界间的差距。
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