Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely $n^4/64+O(n^3)$, thus matching asymptotically the conjectured value of $\text{cr}(K_n)$. Let $\text{cr}_P(G)$ denote the crossing number of a graph $G$ in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of $K_n$ is $(n^4/8π^2)+O(n^3)$. In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if $\lim_{n\to\infty} \text{cr}_P(K_n)/n^4=1/8π^2$. We construct drawings of $K_n$ in the projective plane that disprove this.
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.
We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.
We study general Delaunay-graphs, which are natural generalizations of Delaunay triangulations to arbitrary families, in particular to pseudo-disks. We prove that for any finite pseudo-disk family and point set, there is a plane drawing of their Delaunay-graph such that every edge lies inside every pseudo-disk that contains its endpoints.
This article studies a generalization of magic squares to finite projective planes. In traditional magic squares the entries come from the natural numbers. This does not work for finite projective planes, so we instead use Abelian groups. For each finite projective plane we demonstrate a small group over which the plane can labeled magically. In the prime order case we classify all groups over which the projective plane can be made magic.
This paper studies the existence of finite non-Desarguesian flag-transitive projective plane, giving necessary conditions in terms of polynomial equations over finite fields of characteristic $3$. This sheds light on the longstanding conjecture that every finite flag-transitive projective plane is Desarguesian.
Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.
We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds fo…
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique way in a projective plane of order 16.
We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.
We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Grötzsch class, received applying Mycielski's Construction to odd cycle.
We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.
We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power $q$, find a minimal non-orientable submatroid of the projective plane over the $q$-element field.
Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.
We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs.
We determine the partition dimension of the incidence graph $G(Π_q)$ of the projective plane $Π_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(Π_q))\leq (4+o(1))\log_2{q}.$
In this study we performed a computer search for unitals in planes of order 16. Some new unitals were found and we show that some unitals can be embedded in two or more different planes.
We show that if a planar graph $G$ has a plane straight-line drawing in which a subset $S$ of its vertices are collinear, then for any set of points, $X$, in the plane with $|X|=|S|$, there is a plane straight-line drawing of $G$ in which the vertices in $S$ are mapped to the points in $X$. This solves an open problem posed by Ravsky and Verbitsky in 2008. In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings.
A combinatorial characterization of resolvable Steiner 2-$(v,k,1)$ designs embeddable as maximal arcs in a projective plane of order $(v-k)/(k-1)$ is proved, and a generalization of a conjecture by Andries Brouwer \cite{Br} is formulated.
The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a m…
For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$ …
We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University（应用数学系，数学物理学院，查理大学）；Computer Science Institute, Faculty of Mathematics and Physics, Charles University（计算机科学研究所，数学物理学院，查理大学）
We prove that the entropy compression method systematized by L. Esperet and A. Parreau [11] can be applied to any problem formulated in the variable version of the Lovász Local Lemma. As an application, we prove the existence of legitimate colorings for projective planes with small orders, which extends results of N. Alon and Z. Füredi [2]. In fact, we allow different numbers of colors, proving that projective planes of any order can be legitimate colored with 42 colors.
An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as…
In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length least $9$.
Every simple drawing of a graph in the plane naturally induces a rotation system, but it is easy to exhibit a rotation system that does not arise from a simple drawing in the plane. We extend this to all surfaces: for every fixed surface $Σ$, there is a rotation system that does not arise from a simple drawing in $Σ$.
Every finite graph admits a \emph{simple (topological) drawing}, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, \emph{$k$-planar graphs} are those graphs that can be drawn so that every edge has at most $k$ crossings (i.e., they admit a \emph{$k$-plane drawing}). It is known that for $k\le 3$, every $k$-planar graph admits a $k$-plane simple drawing. But for $k\ge 4$, there exist $k$-planar graphs that do not admit a $k$-plane simple drawing. Answering a questi…
Simple drawings are drawings of graphs in which two edges have at most one common point (either a common endpoint, or a proper crossing). It has been an open question whether every simple drawing of a complete bipartite graph $K_{m,n}$ contains a plane spanning tree as a subdrawing. We answer this question to the positive by showing that for every simple drawing of $K_{m,n}$ and for every vertex $v$ in that drawing, the drawing contains a shooting star rooted at $v$, that is, a plane spanning tree containing all edges incident to $v$.
We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives hope to extend it to other surfaces. Moreover, our approach can be used to provide an efficient algorithm turning a Hanani-Tutte drawing on the projective plane into an embedding.
This report gives an overview of the history of finite projective planes and their properties before going on to outline the proof that no projective plane of order 10 exists. The report also investigates the search carried out by MacWilliams, Sloane and Thompson in 1970 [12] and confirms their result by providing independent verification that there is no vector of weight 15 in the code generated by the projective plane of order 10.
We study Poncelet's Theorem in the four non-isomorphic finite projective planes of order 9. Among these planes, only the Desarguesian plane turns out to be a Poncelet plane, while the other three planes which are constructed over the miniquaternion near-field of order 9, are not. This gives a complete discussion of Poncelet's Theorem in finite projective planes of order 9.
I adapt Berlekamp's light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension.
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a mod2-valued self-intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments accessible to mathematicians not specialized in any of the areas discussed. So most part of this survey could be studied before textbooks on algebraic topology, as an introduction to starting idea…
A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the the points of a non-Desarguesianprojective plane must not contain any components.
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type $\widetilde{A}_2$ and respective absolute type $\widetilde{\mathsf{A}}_2$, $\widetilde{\mathsf{A}}_5$ and $\widetilde{\mathsf{E}}_6$.
A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.
The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.
In this paper we study the area requirements of planar greedy drawings of triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family $\cal H$ of subdivisions of triconnected plane graphs and claimed that every planar greedy drawing of the graphs in $\mathcal H$ respecting the prescribed plane embedding requires exponential area. However, we show that every $n$-vertex graph in $\cal H$ actually has a planar greedy drawing respecting the prescribed plane embedding on an $O(n)\times O(n)$ grid. This reopens the question whether triconnected planar graphs admit planar greedy drawings o…
Suppose that a group $G$ acts transitively on the points of $\mathcal{P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.
Charles University（查理大学）；Università degli Studi di Perugia（佩鲁贾大学）；University of Warsaw（华沙大学）；Universität Passau（帕绍大学）；Czech Technical University in Prague（捷克理工大学）；Universität Würzburg（维尔茨堡大学）；Technische Universität München（慕尼黑工业大学）
Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.
Recently, the authors of the present work (together with M. N. Kolountzakis) introduced a new version of the non-commutative Delsarte scheme and applied it to the problem of mutually unbiased bases. Here we use this method to investigate the existence of a finite projective plane of a given order d. In particular, a short new proof is obtained for the nonexistence of a projective plane of order 6. For higher orders like 10 and 12, the method is non decisive but could turn out to give important supplementary informations.
