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.