A \(KS_2(v;1,\lambda)\) is called indecomposable if it is not isomorphic to the direct sum of a \(KS_2(v;1,\lambda_1)\) with a \(KS_2(v ;1,\lambda_2)\) for some \(\lambda_1\) and \(\lambda_2\) which add to \(\lambda\). In this note, we show that there exists an indecomposable \(KS_2(v;1,\lambda)\) for \(v \equiv 0 \pmod{2}\), \(v \geq 4\), and \(\lambda \geq 2\).

A graph \(G\) is said to be \(m\)-neighbour-connected if the neighbour-connectivity of the graph, \(K(G) = m\). A graph \(G\) is said to be critically \(m\)-neighbour-connected if it is \(m\)-neighbour-connected and the removal of the closed neighbourhood of any one vertex yields an \((m-1)\)-neighbour-connected subgraph. In this paper, we give some upper bounds of the minimum size of the critically \(m\)-neighbour-connected graphs of any fixed order \(v\), and show that the number of edges in a minimum critically \(m\)-neighbour-connected graph with order \(v\) (a multiple of \(m\)) is \(\left\lceil\frac{1}{2}mv\right\rceil\).

We classify the finite partially ordered sets which satisfy certain homogeneity conditions. One of the conditions considered is that the automorphism group of the partially ordered set acts multiply transitively on the set of elements of the same height.

Let \(G = (V, E)\) be a graph or digraph, and let \(r\) and \(s\) be two positive integers. A subset \(U\) of \(V\) is called an \((r, s)\) dominating set if for any \(v \in V – U\), there exists \(u \in U\) such that \(d(u,v) \leq r\) and for any \(u \in U\) there exists \(u’ \in U\) (\(u’ \neq u\)) for which \(d(u’,u) \leq s\). For graphs, a \((1,1)\)-dominating set is the same as a total dominating set. The \((r, s)\)-domination number \(\delta_{r,s}(G)\) of a graph or digraph \(G\) is the cardinality of a smallest \((r,s)\)-dominating set of \(G\). Various bounds on \(\delta_{r,s}(G)\) are established including that, for an arbitrary connected graph of order \(n \geq 2\), if \(s \leq r+1\) then \(\delta_{r,s}(G) \leq \max\left(\frac{2n}{r+s+1},2\right)\), and if \(s \geq r+1\) then \(\delta_{r,s}(G) < \max\left(\frac{n}{r+1},2\right)\). Both bounds are sharp.

Adjusted orthogonal row-column designs have certain desirable properties. In this paper we give a definition of adjusted orthogonal row-column designs, summarise the known designs, give some construction methods and indicate some open problems. We briefly consider the relationship between adjusted orthogonal row-column designs and orthogonal main effects block designs.

The Pfaffian of the symbols \(a_{ij}\) with \(i<j\) has a combinatorial interpretation as the signed weight generating function of perfect matchings in the complete graph. By properly specializing the variables, this generating function reduces to the signed weight generating function for the perfect matchings in an arbitrary simple graph. We construct a weight and sign preserving bijection between two appropriately constructed spaces of permutations: permutations with even cycles and pairs of involutions without fixed points. This bijection gives a purely combinatorial proof that the determinant of a zero axial skew-symmetric matrix is equal to the square of the Pfaffian.

We construct nilpotent \(SQS\)-skeins of class \(n\), for any positive integer \(n\). These \(SQS\)-skeins are all subdirectly irreducible algebras. The nilpotent \(SQS\)-skeins of class \(n\), which are constructed in this paper, are also solvable of order \( \leq \frac{n+1}{2} \) if \(n\) is odd, and of order \(\leq 1+\frac{1}{2}n\) if \(n\) is even.

We employ a well-known class of designs to give a complete solution to the problem of determining the spectrum of uniform semiframes with block size two. As a corollary we prove that the complete graph \(K_{gu}\), admits a one-factorization with an orthogonal set of \(u\) disjoint sub-one-factorizations of \(K_g\) if and only if \(g\) is even and \(u\geq3\).

This paper concerns the existence of graphs and digraphs with prescribed mean distance and the existence of graphs with prescribed mean local connectivity.

Let \(v\), \(k\), and \(\lambda\) be positive integers. A \((v, k, \lambda)\)-Mendelsohn design (briefly \((v,k,\lambda)\)-MD) is a pair \((X,B)\) where \(X\) is a \(v\)-set (of points) and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair of points of \(X\) are consecutive in exactly \(\lambda\) blocks of \(B\). A set of $\delta$ distinct elements \(\{a_1,a_2,…,a_\delta\}\) is said to be cyclically ordered by \(a_1<a_2<…<a_k<a_1\) and the pair \(a_i,a_{i+t}\) are said to be \(t\)-apart in a cyclic \(k\)-tuple \((a_1,a_2,…,a_k)\) where \(i+ t\) is taken modulo \(k\). If for all \(t=1,2,…, k-1\), every ordered pair of points of \(X\) are \(t\)-apart in exactly \(\lambda\) blocks of \(B\), then the \((v,k,\lambda)\)-MD is called a perfect design and is denoted briefly by \((v,k,\lambda)\)-PMD. A necessary condition for the existence of a \((v,k,\lambda)\)-PMD is \(\lambda v(v-1)\equiv0\) (mod \(k\)). In this paper, we shall be concerned mainly with the case where \(k=4\). It will be shown that the necessary condition for the existence of a \((v,4,\lambda)\)-PMD, namely, \(\lambda v(v-1)\equiv0\) (mod \(4\)), is also sufficient, except for \(v=4\) and \(\lambda\) odd, \(v=8\) and \(\lambda=1\), and possibly excepting \(v=12\) and \(\lambda=1\). Apart from the existence of a \((12,4,1)\)-PMD, which remains very much in doubt, the problem of existence of \((v,4,\lambda)\)-PMDs is now completely settled.