On quantitative structure of small Ree groups

The main aim of this article is to study quantitative structure of small Ree Groups ²G₂(q). Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order.     

Polychromatic 4-coloring of cubic even embeddings on the projective plane

A polychromatic     k$k$-coloring   of a map G$G$ on a surface is a k$k$-coloring such that each face of G$G$ has all k$k$ colors on its boundary vertices. An even embedding     G$G$ on a surface is a map of a simple graph on the surface such that each face of G$G$ is bounded by a cycle of even length. In this paper, we shall prove that a cubic even embedding G$G$ on the projective plane has a polychromatic proper 4-coloring if and only if G$G$ is not isomorphic to a Möbius ladder with an odd number of rungs. For proving the theorem, we establish a generating theorem for 3-connected Eulerian multi-triangulations on the projective plane.     

Minimal non-orientable matroids in a projective plane

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.     

On the dimensions of the binary codes of a class of unitals

Let U_β be the special Buekenhout-Metz unital in PG(2,q²), formed by a union of q conics, where q=p^e is an odd prime power. It can be shown that the dimension of the binary code of the corresponding unital design U_β is less than or equal to q³+1−q. Baker and Wantz conjectured that equality holds. We prove that the aforementioned dimension is greater than or equal to .     

Enumeration of maps on the projective plane

1. IntroductionA lnap is rooted if an edge is distinguished togetl1er with an end and a side of the edge.An edge belo11ging to only one face is called double (or 8ingular by some author), al1 othersbelonging to exactly two faces are called s1ngle. The enumeration of rooted p1anar maps wasfirst introduced by Tutte['], Techniques originated by Tutte [2,3l for enumerating variousclasses of rooted Inaps on tIle sphere are here applied to the c1asses of alI rooted maps onthe projective plane. Th…     

1-Embeddability of complete multipartite graphs on the projective plane

A graph is 1-embeddable on a closed surface if there exists a drawing of the graph on the surface such that each edge crosses at most one other edge at a point. In this paper, we determine all the 1-embeddable complete k-partite graphs on the projective plane.     

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997     

LDPC codes generated by conics in the classical projective plane

We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Enumeration of the doubles of the projective plane of order 4

A classification of the doubles of the projective plane of order 4 with respect to the order of the automorphism group is presented and it is established that, up to isomorphism, there are 1 746 461 307 doubles. We start with the designs possessing non-trivial automorphisms. Since the designs with automorphisms of odd prime orders have been constructed previously, we are left with the construction of the designs with automorphisms of order 2. Moreover, we establish that a 2-(21,5,2) design cannot be reducible in two inequivalent ways. This makes it possible to calculate the number of designs with only the trivial automorphism, and consequently the number of all double designs. Most of the computer results are obtained by two different approaches and implementations.     

Group-labeled light dual multinets in the projective plane

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 of $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length at least 9.     

Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph

We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ).     

A lower bound for the minimum weight of the dual 7-ary code of a projective plane of order 49

Existing bounds on the minimum weight d ^⊥ of the dual 7-ary code of a projective plane of order 49 show that this must be in the range 76 ≤ d ^⊥ ≤ 98. We use combinatorial arguments to improve this range to 88 ≤ d ^⊥ ≤ 98, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual codes of finite projective planes is also included. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Block Primitive 2-(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,1) Designs Admitting a Ree Group of Characteristic Two

Let be a 2-(v,k,1) design, and let G be a group of automorphisms of . We show that if G is block primitive, then G does not admit a Ree group as its socle.