A basis is exhibited for the first homology space of a surface over a field. This basis is found by extending a basis of the boundary cycle space of an embedded graph to the cycle space of the graph.

Some interesting implications of the multiplier conjecture are pointed out in this paper. We show the nonexistence of seven unknown difference sets, assuming the multiplier conjecture. If any of those difference sets is found by other means, it would, therefore, disprove the multiplier conjecture. These difference sets correspond to seven missing entries in Lander’s table.

Groups \(\&\) Graphs is a research tool for computing with graphs and their automorphism groups. This note describes the various kinds of information that it can provide.

We show that there are \(1281\) non-isomorphic residual \((16, 24, 9, 6, 3)\)-designs.

The cycle rank, \(r(G)\), of a graph \(G = (V, E)\) is given by \(r(G) = |E| – |V| + 1\). Let \(f(k, r)\) be the minimum number of cycles possible in a \(k\)-connected graph with cycle rank \(r\). We show \(f(1, r) = r\), \(f(2, r) = \binom{r+1}{2}\), \(f(3, r) = r^2 – r + 1\) and characterize the extremal graphs. Bounds are obtained for \(f(k, r)\), \(k \geq 4\); the upper bound is polynomial in \(r\).

We prove that for any odd positive integer \(n > 1\) and for any sufficiently large integer \(v > v_0(n)\), there exists a Nested Steiner \(n\)-Cycle System of order \(v\) if and only if \(v \equiv 1 \pmod{2n}\). This gives rise to many new classes of perpendicular arrays.

In this paper, we examine the concept of cohesion, which was first introduced in \([2]\) and further studied in \([5]\). Our purpose is to consider the global effects on cohesion when an edge is deleted from a given graph. The earlier paper dealt with such effects when an edge was added, and then in a local sense. After some preliminary discussions and definitions, we move on to display graphs that are “nearly stable” under edge deletion and to further discover an infinite class of \(2\)-connected graphs that are indeed “stable”. This result is followed by some discussion of graphs that have more than one block.

Let \(V\) be a set of \(v\) elements. Let \(G_1, G_2, \ldots, G_m\) be a partition of \(V\) into \(m\) sets. A \(\{G_1, G_2, \ldots, G_m\}\)-frame \(F\) with block size \(k\), index \(\lambda\) and latinicity \(\mu\) is a square array of side \(v\) which satisfies the properties listed below. We index the rows and columns of \(F\) with the elements of \(V\). (1) Each cell is either empty or contains a \(k\)-subset of \(V\). (2) Let \(F_i\) be the subsquare of \(F\) indexed by the elements of \(G_i\). \(F_i\) is empty for \(i = 1, 2, \ldots, m\). (3) Let \(j \in G_i\). Row \(j\) of \(F\) contains each element of \(V – G_i\) \(\mu\) times and column \(j\) of \(F\) contains each element of \(V – G_i\) \(\mu\) times. (4) The collection of blocks obtained from the nonempty cells of \(F\) is a \(GDD(v; k; G_1, G_2, \ldots, G_m; 0, \lambda)\). If \(|G_i| = h\) for \(i = 1, 2, \ldots, m\), we call \(F\) a \((\mu, \lambda, k, m, h)\)-frame.
Frames with \(\mu=\lambda=1\) and \(k = 2\) were used by D.R. Stinson to establish the existence of skew Room squares and Howell designs. \((1, 2; 3, m, h)\)-frames with \(h = 1, 3\) and \(6\) have been studied and can be used to produce \(KS_3(v; 1, 2)s\). In this paper, we prove the existence of \((2, 4; 3, m, h)\)-frames for \(h = 3\) and \(6\) with a finite number of possible exceptions. We also show the existence of \((2, 4; 3, m, 1)\)-frames for \(m \equiv 1 \pmod{12}\). These frames can be used to construct \(KS_3(v; 2, 4)s\).

We give a brief account of some recent results on edge-colouring simple graphs and of some recent results on the total-chromatic number of simple graphs. We illustrate the kind of arguments which have been found to be successful by proving one of the simpler results on edge-colouring graphs, and by showing how to apply this to obtain one of the recent results on the total-chromatic number.

The question of whether all \(B[k,t;k^2]\) designs are \(t\)-resolvable is answered in the affirmative for \(k=3\) and \(t=3\), when the design has no repeated blocks. It is further shown that all such \(B[3,3;9]\) designs are also \(2\)-resolvable.

A binary code has bounded synchronization delay if there exists an integer \(s\) such that at most \(s\) consecutive bits are required to establish word synchronization in any message. The code whose words are lexicographically least in the non-periodic orbits determined by cyclic permutation of all words of length \(n\) is called the canonical bounded synchronization delay code. It has the maximal number of words possible in a synchronizable code of fixed word length. Any code of fixed word length \(n\) can be represented as a set of vertices in the \(n\)-cube. We prove that the canonical bounded synchronization delay code is a connected subset of the \(n\)-cube.

A degree sequence which is realisable by a \(3\)-uniform hypergraph is called \(3\)-graphic. Seven necessary conditions, one sufficient condition, and one equivalent condition for degree sequences to be \(3\)-graphic are derived. Moreover, four special classes of degree sequences are examined. For each class an equivalent condition for the sequences in this class to be \(3\)-graphic is derived. Using these conditions, all the \(3\)-graphic degree sequences of length at most seven have been determined.

We prove that if \(m\) is even then a partial \(m\)-cycle system on \(n\) vertices can be embedded in an \(m\)-cycle system on \(2mn+1\) vertices.

Ideas are described that speed up the lattice basis reduction algorithm of Lenstra, Lenstra, and Lovász \([11]\) in practice. The resulting lattice basis reduction algorithm reduces the multiprecision operations needed in previous approaches. This paper describes these ideas in detail for lattices of the particular form arising from the subset sum (exact knapsack) problem. The idea of applying the \(L^3\) algorithm to the subset sum problem is due to Lagarias and Odlyzko \([8]\). The algorithm of this paper also uses a direct search for short vectors simultaneously with the basis reduction algorithm. Extensive computational tests show that this algorithm solves, with high probability, instances of low-density subset sum problems and has two major advantages over the method of Lagarias and Odlyzko: running time is an order of magnitude smaller and higher-density subset sum problems are solved.

It is shown that the collection of all \(\binom{9}{4}\) distinct quadruples chosen from a set of nine points can be partitioned into nine mutually disjoint \(3-(8,4,1)\) designs in just two non-isomorphic ways. Two proofs of this result are given: one by direct construction, the other by extending sets of eight mutually disjoint \(2-(7,3,1)\) designs based on a set of eight points.

Suppose that one is given \(v\) elements, and wishes to form a design that covers all \(t\)-sets from these elements exactly once. The design is to obey the further restriction that the longest block in the design has \(k\) elements in it; furthermore, we wish the design to contain as few blocks as possible.

The number of blocks in such a minimal design is denoted by the symbol \(\text{g}^{(\text{k})}(1,t;v)\); determination of this number is closely connected with the behaviour of Steiner Systems. Recently, much progress has been made in two important cases, namely, when \(t = 2\) (pairwise balanced designs) and \(t = 3\) (designs with balance on triples). This survey paper presents the subject from its inception up to current results.

Recent examples of perfect 1-factorizations arising from dicyclic groups have led to the question of whether or not dicyclic groups have symmetric sequencings. For every positive integer \(n\geq2\), there is a dicyclic group of order \(4n\). It is known that if \(n \geq 3\) is odd, then the dicyclic group of order \(4n\) has a symmetric sequencing. In this paper, a new proof is given for the odd case; a consequence being that in this situation sequencings abound. A generalization of the original proof is exploited to show that if \(n\geq 4\) is even and is not twice an odd number, then the dicyclic group of order \(4n\) has a symmetric sequencing.