The present paper continues the study (begun by Quattrocchi, Colouring 4-cycle systems with specified block colour pattern: the case of embedding P3-designs, Electron. J. Combin., 8 (2001)) of 4-cycle systems with specified block colour patterns that are also strict colourings in the sense of Voloshin (Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, American Mathematical Societ...

This article is a contribution to the study of the automorphism groups of 2 − (v, k, 1) designs. Let G act as a block-transitive and point-primitive automorphism group of a non-trivial design D with v points and blocks of size k. Set k2 = (k, v−1). Assume q = pf for some prime p and positive integer f . If q ≥ [(k2k − k2 + 1)f ]2, then Soc(G), the socle of G, is not PSL(2, q). Mathematics Subje...

Let C be a C4-design of order n and index λ, on the vertex set V , |V | = n. If V1∪· · ·∪Vm = V is a partition of the vertex set, such that the intersections of the C ∈ C with Vi form a P3-design of order |Vi| and the same index λ, for each 1 ≤ i ≤ m, then 2 ≤ m ≤ log3(2n+1). The minimum bound is best possible for every λ. The maximum bound is best possible for λ = 2, and hence also for every e...

In this paper, we consider the projective special linear group $PSL_2(59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2(59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2(59)$ and $PSL_2(59):2$ appear as the automorphism group of the constructed designs.

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute...

Some binary matrices like (1,-1) and (1,0) were studied by many authors like Cohn [3], Wang [31], Ehlich [5] and Ehlich and Zeller [6], and Mohan, Kageyama, Lee, and Gao in [18]. In this recent paper by Mohan et al considered the M-matrices of Type I and II by studying some of their properties and applications. In the present paper we discuss the M-matrices of Type III, and study their properti...

The Mn-matrix was defined by Mohan [20] in which he has shown a method of constructing (1,-1)-matrices and studied some of their properties. The (1,-1)-matrices were constructed and studied by Cohn [5],Wang [33], Ehrlich [8] and Ehrlich and Zeller[9]. But in this paper, while giving some resemblances of this matrix with Hadamard matrix, and by naming it as M-matrix, we show how to construct par...

An Sq[t,k,v] q-Steiner system is a collection of k-dimensional subspaces of the v-dimensional vector space Fq over the finite field Fq with q elements, called blocks, such that each t-dimensional subspace of Fq is contained in exactly one block. The smallest admissible parameters for which a q-Steiner system could exist is S2[2,3,7]. Up to now the issue whether q-Steiner systems with these para...

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point imprimitive group action, for a given value of gcd(k, r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd...

Let D be the triangle with an attached edge (i. e. D is the “kite”, a graph having vertices {a0, a1, a2, a3} and edges {a0, a1}, {a0, a2}, {a1, a2}, {a0, a3}). Bermond and Schönheim [6] proved that a kite-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial kite-design of order n ≥ 8, and let V ⊂ W with |V | = v < n. A path design (V,P) of order v and block siz...