Partial ‐Parallelisms in Finite Projective Spaces

In this paper, we consider the following question. What is the maximum number of pairwise disjoint k‐spreads that exist in PG()? We prove that if divides and then there exist at least two disjoint k‐spreads in PG() and there exist at least pairwise disjoint k‐spreads in PG(n, 2). We also extend the known results on parallelism in a projective geometry from which the points of a given subspace were removed.     

具有逆断面的纯正群的构造

本文用具有半格断面的带和Clifford半群给出具有逆断面的纯正群的一个构造定理.     

Approximate min–max relations for odd cycles in planar graphs

We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss recently proved that this ratio is at most 2; we also give a short proof of their result. This work was supported by FNRS, NSERC (PGS Master award, Canada Research Chair in Graph Theory, award 288334-04) and FQRNT (award 2005-NC-98649).     

On the number of minimal transversals in 3-uniform hypergraphs

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most cⁿ, where c≈1.6702. The best known lower bound for this number, due to Tomescu, is adⁿ, where d=10^1/5≈1.5849 and a is a constant.     

The Existence of Mixed Orthogonal Arrays with Four and Five Factors of Strength Two

Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception.     

A Non-expensive Homological Helly Theorem

We will discuss the following result: for a topological space X with the property that $H_k(U)=0$ for $k⩾d$ and every open subset U of X, a finite family of open sets in X has nonempty intersection if for any subfamily of size j, $1⩽j⩽d+1$, the ($d-j$)-dimensional reduced homology group of its intersection is zero. We also use this theorem to discuss new results concerning transversal affine planes to families of convex sets.     

Minimum size transversals in uniform hypergraphs

A set of vertices in a hypergraph which meets all the edges is called a transversal. The transversal number τ(H)$\tau \left(H\right)$ of a hypergraph H$H$ is the minimum cardinality of a transversal in H$H$. A classical greedy algorithm for constructing a transversal of small size selects in each step a vertex which has the largest degree in the hypergraph formed by the edges not met yet. The analysis of this algorithm (by Chvátal and McDiarmid (1992)  [3]) gave some upper bounds for τ(H)$\tau \left(H\right)$ in a uniform hypergraph H$H$ with a given number of vertices and edges. We discuss a variation of this greedy algorithm. Analyzing this new algorithm, we obtain upper bounds for τ(H)$\tau \left(H\right)$ which improve the bounds by Chvátal and McDiarmid.     

On Inverse Transversals of Ordered Regular Semigroups

We first consider an ordered regular semigroup S in which every element has a biggest inverse and determine necessary and sufficient conditions for the subset S ^○ of biggest inverses to be an inverse transversal of S. Such an inverse transversal is necessarily weakly multiplicative. We then investigate principally ordered regular semigroups S with the property that S ^○ is an inverse transversal. In such a semigroup we determine precisely when the set S ^☆ of biggest pre-inverses is a subsemigroup and show that in this case S ^☆ is itself an inverse transversal of a subsemigroup of S. The ordered regular semigroup of 2 × 2 boolean matrices provides an informative illustrative example. The structure of S, when S ^☆ is a group, is also described.     

On Generalized Orthodox Transversals

Generalized orthodox transversals are introduced in this article, and some characterizations associated with them are obtained. The related results of Chen on quasi-ideal orthodox transversals obtained in 1999 and Zhang and Wang on generalized inverse transversals obtained in 2008 are generalized and amplified. A structure theorem of a regular semigroup with a perfect generalized orthodox transversal is also established.