Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

A series of balanced incomplete block (BIB) designs with parameters , is constructed, where , w any integer.

An isomorphic factorisation of a digraph D is a partition of its arcs into mutually isomorphic subgraphs. If such a factorisation of D into exactly t parts exists, then t must divide the number of arcs in D. This is called the divisibility condition. It is shown conversely that the divisibility condition ensures the existence of an isomorphic factorisation into t parts in the case of any complete digraph. The sufficiency of the divisibility condition is also investigated for complete m-partite digraphs. It is shown to suffice when m = 2 and t is odd, but counterexamples are provided when m = 2 and t is even, and when m = 3 and either t = 2 or t is odd.

Grant (1976) has attempted to establish a relationship between fixing subgraphs and smoothly embeddable subgraphs. Here we give counterexamples to his two main lemmas and two characterizing theorems. We then go on to give our own version of these lemmas and theorems.

In this paper we prove two analogues of the following theorem:

If is a collection of distinct subsets of {1, …, m} such that , i ≠ j ⇒ Ai ∩ Aj ≠ ∅, Ai ∪ Aj ≠ {1, …, m} then .

The structure is determined for the existence of some amicable weighing matrices. This is then used to prove the existence and non-existence of some amicable orthogonal designs in powers of two.

An asymptotic expansion is obtained for this sequence, of interest in combinatorial analysis. Values are given for the constants appearing in the leading term and a numerical comparison made.

An equidistant permutation array (EPA) is a ν × r array defined on an r-set, R, such that (i) each row is a permutation of the elements of R and (ii) any two distinct rows agree in λ positions (that is, the Hamming distance is (r−λ)).

Such an array is said to have order ν. In this paper we give several recursive constructions for EPA's.

The first construction uses a resolvable regular pairwise balanced design of order v to construct an EPA of order ν. The second construction is a generalization of the direct product construction for Room squares.

We also give a construction for intersection permutation arrays, which arrays are a generalization of EPA's.

For some integer modulus q let F be a subset of Z(q), the additive group of residues modulo q. As in «1» we shall write

and

The letters a, b, n, m, t (perhaps with suffixes) always denote natural numbers. A, B, S denote finite sets of natural numbers. |A| stands for the cardinality of A.

For a given constant c > 1 we say that the set A has property α(c) if there are at most c|A| differences a − b ≥ 0 for a, b ∈ A.

A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K(G) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ(G) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph Kp we have

Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n–sphere in En+1, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (–1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n–1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 …, Pr. The empty set is the only (–1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 …, Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for all .

By a plane tree or rooted plane tree is meant a realization of a tree or rooted tree by points and arcs in the plane. By an isomorphism between two plane trees or rooted plane trees is meant an isomorphism in the usual sense for such trees which preserves the clockwise cyclic order of the edges about each node. In [2] Harary, Prins, and Tutte demonstrate how Polya's Theorem may be used to obtain formal expressions for the enumerating functions for unrooted, rooted, and other species of plane trees. In the process they obtain the explicit formula

for the number of nonisomorphic planted plane trees with n ≥ 1 edges. (A planted tree is a rooted tree with root at a node of degree one.)

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems.

Theorem A. Let En be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized.

Theorem B. Let En be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set.

Here a distance d is realized within a set S, if there are points x, y in S at distance d apart.

A tournament is a relational structure on the non-empty set T such that for x, y ∈ T exactly one of the three relations

holds. Here x → y expresses the fact that {x, y} ∈ → and we sometimes write this in the alternative form y ← x. Extending the notation to subsets of T we write A → B or B ← A if a → b holds for all pairs a, b with a ∈ A and b ∈ B. is a subtournament of , and is an extension of , if T′ ⊂ T and →′ is the restriction of → to T′; we will usually write 〈′, → 〉 instead of 〈 ′, → ′〉. In particular, if |T − T′| = k, we call a k-poinf extension of .

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

Trees are basic in graph theory and its applications to many fields, such as chemistry, electric network theory, and the theory of games. König [7; 47-48] gives an interesting historical account of independent discoveries of trees by Kirchhoff, Cayley, Sylvester, Jordan, and others who were working in a variety of fields.

There are many equivalent ways of defining trees, the most common being: A tree is a graph which is connected and has no circuits. Figure 1 shows the trees with up to six vertices; those with up to 10 vertices can be found in Harary [5; 233–234]. Some other possible definitions of a tree are the following: (1) A tree is a graph which is connected and has one more vertex than edge. (2) A tree is a graph which has no circuits and has one more vertex than edge. For these and other characterizations see Berge [4; 152ff.] and Harary [5; 32ff.]. A less common definition is by induction: The graph consisting of a single vertex is a tree, and a tree with n + 1 vertices is obtained from a tree with n vertices by adding a new vertex adjacent to exactly one other vertex.

Plummer (see [2; p. 69]) conjectured that the square of every block is hamiltonian, and this has just been proved by Fleischner [1]. It was shown by Karaganis [3] that the cube of every connected graph, and hence the cube of every tree, is hamiltonian. Our present object is to characterize those trees whose square is hamiltonian in three equivalent ways.

We follow the terminology and notation of the book [2]. In particular, the following concepts are used in stating our main result. A graph is hamiltonian if it has a cycle containing all its points. The graph with the same points as G, in which two points are adjacent if their distance in G is at most 2, is denoted by G2 and is called the square of G. The subdivision graph S(G) is formed (Figure 1) by inserting a point of degree two on each line of G.

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.