The minimal number of triples required to represent all quintuples on an \(n\)-element set is determined for \(n \leq 13\) and all extremal constructions are found. In particular, we establish that there is a unique minimal system on 13 points, namely the 52 collinear triples of the projective plane of order 3.

A set \(T\) with a binary operation \(+\) is called an operation set and denoted as \((T, +)\). An operation set \((S, +)\) is called \(q\)-free if \(qx \notin S\) for all \(x \in S\). Let \(\psi_q(T)\) be the maximum possible cardinality of a \(q\)-free operation subset \((S, +)\) of \((T, +)\).

We obtain an algorithm for finding \(\psi_q({N}_n)\), \(\psi_q({Z}_n)\) and \(\psi_q(D_n)\), \(q \in {N}\), where \({N}_n = \{1, 2, \ldots, n\}\), \(( {Z}_n, +_n)\) is the group of integers under addition modulo \(n\) and \((D_n, +_n)\) is the dihedral group of order \(2n\).

A decomposition of \(K_v\) into \(2\)-perfect \(8\)-cycles is shown to exist if and only if \(v \equiv 1 (\mod 16\)).

The binary matroids with no three- and four-wheel minors were characterized by Brylawski and Oxley, respectively. The importance of these results is that, in a version of Seymour’s Splitter Theorem, Coullard showed that the three- and four-wheel matroids are the basic building blocks of the class of binary matroids. This paper determines the structure of a class of binary matroids which almost have no four-wheel minor. This class consists of matroids \(M\) having a four-wheel minor and an element \(e\) such that both the deletion and contraction of \(e\) from \(M\) have no four-wheel minor.

A pairwise balanced design (PBD) of index \(I\) is a pair \((V,{A})\) where \(V\) is a finite set of points and \(A\) is a set of subsets (called blocks) of \(V\), each of cardinality at least two, such that every pair of distinct points of \(V\) is contained in exactly one block of \(A\). We may further restrict this definition to allow precisely one block of a given size, and in this case the design is called a PBD \((\{K, k^*\},v)\) where \(k\) is the unique block size, \(K\) is the set of other allowable block sizes, and \(v\) is the number of points in the design.

It is shown here that a PBD \((\{5, 9^*\},v)\) exists for all \(v \equiv 9\) or 17 mod 20, \(v \geq 37\), with the possible exception of \(49\), and that a PBD \((\{5, 13^*\},v)\) exists for all \(v \equiv 13 \mod 20\), \(v \geq 53\).

A partition \(\mathcal{D} = \{V_1, \ldots, V_m\}\) of the vertex set \(V(G)\) of a graph \(G\) is said to be a star decomposition if each \(V_i\) (\(1 \leq i \leq m\)) induces a star of order at least two.
In this note, we prove that a connected graph \(G\) has a star decomposition if and only if \(G\) has a block which is not a complete graph of odd order.

This note recapitulates the definition of a ‘double Youden rectangle’, which is a particular kind of balanced Graeco-Latin design obtainable by superimposing a second set of treatments on a Youden square, and reports the discovery of examples that are of size \(8 \times 1\). The method by which the examples were obtained seems likely to be fruitful for the construction of double Youden rectangles of larger sizes.

It has been shown that there exists a resolvable spouse-avoiding mixed-doubles round robin tournament for any positive integer \(v \neq 2, 3, 6\) with \(27\) possible exceptions. We show that such designs exist for \(19\) of these values and the only values for which the existence is undecided are: \(10, 14, 46, 54, 58, 62, 66\), and \(70\).

A graph \(G\) is homogeneously traceable if for each vertex \(v\) of \(G\) there exists a hamiltonian path in \(G\) with initial vertex \(v\). A graph is called claw-free if it has no induced \(K_3\) as a subgraph.

In this paper, we prove that if \(G\) is a \(k\)-connected (\(k > 1\)) claw-free graph of order \(n\) such that the sum of degrees of any \(k+2\) independent vertices is at least \(n-k\), then \(G\) is homogeneously traceable. For \(k=2\), the bound \(n-k\) is best possible.

As a corollary we obtain that if \(G\) is a \(2\)-connected claw-free graph of order \(n\) such that \(NC(G) \geq (n-3)/2\), where \(NC(G) = \min\{|N(u) \cup N(v)|: uv \notin E(G)\}\), then \(G\) is homogeneously traceable. Moreover, the bound \((n-3)/2\) is best possible.