The Finite Projective Plane and the 5-Uniform Linear Intersecting Hypergraphs with Domination Number Four

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Let Π be a finite projective plane of order n , and let G be a large abelian (or, more generally, quasiregular) collineation group of Π; to be specific, we assume ∣ G ∣ ( n 2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings. Introduction A projective plane is a geometry consisting of points and lines such that any two distinct lines meet in exactly one point, any two distinct points are on exactly one common line, and there are four points no three of which are collinear. Well-known examples are provided by the desarguesian (or classical ) projective planes PG(2, q ), q a prime power, which can be defined over the finite field GF( q ) as follows: points and lines are the 1- and 2-dimensional subspaces of the vector space GF( q ) 3 , and a point is incident with a line if and only if it is a subset of the line. A standard reference for arbitrary projective planes is Hughes and Piper [104]; for the desarguesian case, see Hirschfeld [89].

We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n.

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 present a fairly elementary, self-contained proof of the nonexistence of a finite projective plane of order $6$. Our approach is motivated by theory of binary codes but does not appeal to it directly.

We study a construction of the Mathieu group M 12 using a game reminiscent of Loyd's "15-puzzle." The elements of M 12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M 13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M 12 and M 13. We develop these results, and extend them to the double covers and automorphism groups of M 12 and M 13, using the ternary Golay code and 12 × 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

On the (reverse) cover cost of trees with some given parameters

Due to the large number of light weight, low power devices in the future Internet, an efficient distributed multiple access protocol that can support a small fixed number of machines with continuous traffic along with a large variable number of devices with sporadic traffic is needed. The traffic in the future Internet has fundamentally different requirements: first, the accessing devices are of low power, which limit the transmission range; secondly, the delay is very stringent; thirdly, the number of active devices is time varying. In this paper, we propose a finite projective plane (FPP) based multiple access protocol (FISH) to meet such requirements. A low power node can transmit the replicas of a data packet in multiple time slots to achieve the same range of transmission as a high power node at one time. The initialization time slot of a frame allows for the estimation of the number of active nodes and the adjustment of the size of the frame to maximize the throughput. The interference cancellation and conflict resolution scheme of the FISH protocol is based on the theory of FPP. By using the property that the union of the sets of the FPP sharing one common point is a complete set, we are able to extract each of the individual user signals. One distinguishing feature of the FISH protocol is the amplification of the signal-to-noise ratio (SNR), which can either extend the transmission range or improve the bit error rate without sacrificing the throughput. In the performance evaluation, we demonstrate that the throughput is identical to the Slotted Aloha, while the SNR is magnified m times, where m is the order of the FPP. The proposed FISH protocol is of great use in the machine-to-machine communications as well as traditional cellular and satellite systems.

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 nonpercolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

Let G = (V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> ) be a simple and connected graph. The eccentric connectivity index of G is represented as ξ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> (G) = Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x∈VG</sub> deg <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x)ec <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x), where deg <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x) and ec <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x) represent the degree and the eccentricity of x, respectively. The eccentric adjacency index of G is represented as ξ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad</sup> (G) = Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x∈VG</sub> (S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x)/ec <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (x)), where SQ(x) is the sum of degrees of neighbors of x. In this paper, we determine the trees with the smallest eccentric connectivity index when bipartition size, independence number, and domination number are given. Furthermore, we discuss the trees with the largest eccentric adjacency index when bipartition size, matching number, and independence number are given.

Let G be a (molecular) graph. A perfect matching of G is defined as a set of edges which are independent and cover every vertex of G exactly once. In the article, we present the formula on the number of the perfect matchings of two types of hexagons on the torus by Pfaffians.

ABSTRACTLet be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values for , provided that is not close to the extremal values 0 or . Moreover, using polynomial techniques we show that for every prescribed list of numbers , there exists a point set with the following property: holds for the th line , .

In the late 1950’s, B. Segre introduced the fundamental notion of arcs and complete arcs [48,49]. An arc in a finite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane $${\user1{P}}$$, determining $$n{\left( {\user1{P}} \right)}$$, the size of its smallest complete arc, has been a major open question in finite geometry for several decades. Assume that $${\user1{P}}$$ has order q, it was shown by Lunelli and Sce [41], more than 40 years ago, that $${\left( {\user1{P}} \right)} \geqslant {\sqrt {2q} }$$. Apart from this bound, practically nothing was known about $$n{\left( {\user1{P}} \right)}$$, except for the case $${\user1{P}}$$ is the Galois plane. For this case, the best upper bound, prior to this paper, was O(q3/4) obtained by Szőnyi using the properties of the Galois field GF(q).In this paper, we prove that $$n{\left( {\user1{P}} \right)} \leqslant {\sqrt q }{\kern 1pt} \log^{c} q$$ for any projective plane $${\user1{P}}$$ of order q, where c is a universal constant. Together with Lunelli-Sce’s lower bound, our result determines $$n{\left( {\user1{P}} \right)}$$ up to a polylogarithmic factor. Our proof uses a probabilistic method known as the dynamic random construction or Rodl’s nibble. The proof also gives a quick randomized algorithm which produces a small complete arc with high probability.The key ingredient of our proof is a new concentration result, which applies for non-Lipschitz functions and is of independent interest.

A vertex subset D of G is a dominating set if every vertex in V(G)\D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G)=i(H). Hexagonal systems (2 connected planar graphs whose interior faces are all hexagons) have been extensively studied as they are used to present bezenoid hydrocarbon structures which play an important role in organic chemistry. The domination numbers of hexagonal systems have been studied continuously since 2018 when Hutchinson et al. posted conjectures, generated from a computer program called Conjecturing, related to the domination numbers of hexagonal systems. Very recently in 2021, Bermudo et al. answered all of these conjectures. In this paper, we extend these studies by considering the relationship between the domination number and the independent domination number of hexagonal systems. Although every hexagonal system H with at least two hexagons contains K1,3 as an induced subgraph, we find many classes of hexagonal systems whose domination number is equal to an independent domination number. However, we establish the existence of a hexagonal system H such that γ(H)&lt;i(H) with the prescribed number of hexagons.

Disjoint dominating and 2-dominating sets in graphs