Let G be a graph of order n satisfying that there exists for which every graph of order n and size t is contained in exactly λ distinct subgraphs of the complete graph isomorphic to G. Then G is called t-edge-balanced and λ the index of G. In this article, new examples of 2-edge-balanced graphs are constructed from bipartite graphs and some further methods are introduced to obtain more from old.
We present a construction for minimal blocking sets with respect to -spaces in , the -dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well-known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS-construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to -spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS-constructionthen it is of Rédei-type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.
For each odd , we completely solve the problem of when an m-cycle system of order u can be embedded in an m-cycle system of order v, barring a finite number of possible exceptions. In cases where u is large compared to m, where m is a prime power, or where , the problem is completely resolved. In other cases, the only possible exceptions occur when is small compared to m. This result is proved as a consequence of a more general result that gives necessary and sufficient conditions for the existence of an m-cycle decomposition of a complete graph of order v with a hole of size u in the case where and both hold.
Suppose that and . We construct a Latin square of order n with the following properties:
Hence generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, is what is known as a self-switching Latin square and possesses a near-autoparatopism.
Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- SEEDs are derived for prime powers q and all .
In this paper, we determine the necessary and sufficient conditions for the existence of an equitably ℓ-colorable balanced incomplete block design for any positive integer . In particular, we present a method for constructing nontrivial equitably ℓ-colorable BIBDs and prove that these examples are the only nontrivial examples that exist. We also observe that every equitable ℓ-coloring of a BIBD yields both an equalized ℓ-coloring and a proper 2-coloring of the same BIBD.
This paper examines subsets with at most n points on a line in the projective plane . A lower bound for the size of complete -arcs is established and shown to be a generalisation of a classical result by Barlotti. A sufficient condition ensuring that the trisecants to a complete (k, 3)-arc form a blocking set in the dual plane is provided. Finally, combinatorial arguments are used to show that, for , plane (k, 3)-arcs satisfying a prescribed incidence condition do not attain the best known upper bound.
]]>A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with Turán systems. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.
]]>An is a triple , where X is a set of points, is a partition of X into m disjoint sets of size n and is a set of 4-element transverses of , such that each 3-element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this is called m-cyclic (resp. semi-cyclic). Further, if all block-orbits of an m-cyclic (resp. semi-cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m-cyclic and semi-cyclic , and use them to give new infinite classes of perfect two-dimensional optical orthogonal codes with maximum collision parameter and AM-OPPTS/AM-OPPW property.
A cube design of order v, or a CUBE(v), is a decomposition of all cyclicly oriented quadruples of a v-set into oriented cubes. A CUBE(v) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate.
We show that a nondegenerate CUBE(v) design exists for all integers , and that an unoriented nondegenerate CUBE(v) design exists if and only if and or . A degenerate example of a CUBE(v) design is also given for each integer .
Using the Katona–Kierstead (K–K) definition of a Hamilton cycle in a uniform hypergraph, we investigate the existence of wrapped K–K Hamilton cycle decompositions of the complete bipartite 3-uniform hypergraph and their large sets, settling their existence whenever n is prime.
We detail the enumeration of all two-intersection sets of the five-dimensional projective space over the field of order 3 that are invariant under an element of order 7, which include the examples of Hill (1973) and Gulliver (1996). Up to projective equivalence, there are 6,635 such two-intersection sets.
Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1-factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .
We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.
Let L be a latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 latin subsquares in L can be determined from both and per(L). More generally, we can diagnose from or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic latin squares with equal permanents and equal determinants exist for all orders that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.
We consider the existence problem for a semi-cyclic holey group divisible design of type with block size 3, which is denoted by a 3-SCHGDD of type . When t is odd and or t is doubly even and , the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to two-dimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.
A pseudo-hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that is flag-transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).
We obtained a full computer classification of all complete arcs in the Desarguesian projective plane of order 31 using essentially the same methods as for earlier results for planes of smaller order, i.e., isomorph-free backtracking using canonical augmentation. We tabulate the resulting numbers of complete arcs according to size and automorphism group. We give explicit descriptions for all complete arcs with an automorphism group of size at least 20. In some of these cases the constructions can be generalized to other values of q. In particular, we find arcs of size 20 for any field of order , and a complete 44-arc in PG(2,67) with an automorphism group of order 88. We also correct a result by Kéri : there are 12 complete 22-arcs in PG(2,31) up to projective equivalence, and not 11.
Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any , there exists a CQS for all , and .
Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.
2-(v,k,1) designs admitting a primitive rank 3 automorphism group , where G_{0} belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in [23], are classified.
Intersection numbers for subspace designs are introduced and q-analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q-analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2- subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.
A q-ary code of length n, size M, and minimum distance d is called an code. An code with is said to be maximum distance separable (MDS). Here one-error-correcting () MDS codes are classified for small alphabets. In particular, it is shown that there are unique (5, 5^{3}, 3)_{5} and (5, 7^{3}, 3)_{7} codes and equivalence classes of (5, 8^{3}, 3)_{8} codes. The codes are equivalent to certain pairs of mutually orthogonal Latin cubes of order q, called Graeco-Latin cubes.
A -semiframe of type is a -GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A -SF is a -semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)-SF for any if and only if , , , and .
Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding-Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.