We show that if, for any fixed \(r\), the neighbourhood unions of all \(r\)-sets of vertices are large enough, then \(G\) will have many edge-disjoint perfect matchings. In particular, we show that given fixed positive integers \(r\) and \(c\) and a graph \(G\) of even order \(n\), if the minimum degree is at least \(r + c – 1\) and if the neighbourhood union of each \(r\)-set of vertices is at least \(n/2 + \left(2\lfloor\frac{(c + 1)}{2}\rfloor – 1\right)r\), then \(G\) has \(c\) edge-disjoint perfect matchings, for \(n\) large enough. This extends earlier work by Faudree, Gould and Lesniak on neighbourhood unions of pairs of vertices.

In this paper, necessary and sufficient conditions for a vector to be the fine structure of a balanced ternary design with block size \(3\), index \(3\) and \(\rho_2 = 1\) and \(2\) are determined, with one unresolved case.

Let \(K^d_n\) be the product of \(d\) copies of the complete graph \(K_4\). Wojciechowski [4] proved that for any \(d \geq 2\) the hypercube \(K^d_2\) can be vertex covered with at most \(16\) disjoint snakes. We show that for any odd integer \(n \geq 3\), \(d \geq 2\) the graph \(K^d_n\) can be vertex covered with \(2n^3\) snakes.

Cwatsets are subsets of \(\mathbb{Z}^d_2\) which are nearly subgroups and which naturally appear in statistics and coding theory [8]. Each cwatset can be represented by a highly symmetric hypergraph [7]. We introduce and study the symmetry group of the hypergraph and connect it to the corresponding cwatset. We use this connection to establish structure theorems for several classes of cwatsets.

Bollobás, Brightwell [1] and independently Shi [3] proved the existence of a cycle through all vertices of degree at least \(\frac{n}{2}\) in any \(2\)-connected graph of order \(n\). The aim of this paper is to show that the above degree requirement can be relaxed for \(1\)-tough graphs.

In this paper we investigate the \(k\)th lower multiexponent \(f(n,k)\) for tournament matrices.
It was proved that \(f(m,3) = 2\) if and only if \(m \geq 11\). Thus the conjecture in [2] is disproved. Further we obtain a new sufficient condition for \(f(n,k) = 1\).

The cycle graph \(C(H)\) of a graph \(H\) is the edge intersection graph of all induced chordless cycles of \(H\). We investigate iterates of the mapping \(\overline{C}: G \rightarrow C(\overline{G})\) where \(C\) denotes the map that associates to a graph its cycle graph. We call a graph \(G\) vanishing under \(\overline{C}\) if \(\overline{C^n}(G) = 0\) for some \(n\), otherwise \(G\) is called \(\overline{C}\)-persistent. We call a graph \(G\) expanding under \(\overline{C}\) if \(|\overline{C^n}(G)| \to \infty\) as \(n \to \infty\). We show that the lowest order of a \(\overline{C}\)-expanding graph is \(6\) and determine the behaviour under \(\overline{C}\) of some special graphs, including trees, null graphs, cycles and complete bipartite graphs.

Nonbinary power residue codes are constructed using the relationship between these codes and quasi-cyclic codes. Eleven of these codes exceed the known lower bounds on the maximum possible minimum distance of a linear code.

In this paper the authors study one- and two-dimensional color switching problems by applying methods ranging from linear algebra to parity arguments, invariants, and generating functions. The variety of techniques offers different advantages for addressing the existence and uniqueness of minimal solutions, their characterizations, and lower bounds on their lengths. Useful examples for reducing problems to easier ones and for choosing tools based on simplicity or generality are presented. A novel application of generating functions provides a unifying treatment of all aspects of the problems considered.

Broadcasting refers to the process of information dissemination in a communication network whereby a message is to be sent from a single originator to all members of the network, subject to the restriction that a member may participate in only one message transfer during a given time unit. In this paper we present a family of broadcasting schemes over the odd graphs, \(O_{n+1}\). It is shown that the broadcast time of \(O_{n+1}\), \(b(O_{n+1})\), is bounded by \(2n\). Moreover, the conjecture that \(b(O_{n+1}) = 2n\) is put forward, and several facts supporting this conjecture are given.