If each pair of vertices in a graph \(G\) is connected by a long path, then the cycle space of \(G\) has a basis consisting of long cycles. We propose a conjecture regarding the above relationship. A few results supporting the conjecture are given.

For any tree \(T\), let \(\gamma(T)\) represent the size of a minimum dominating set. Let \({E}_0\) represent the collection of trees with the property that, regardless of the choice of edge \(e\) belonging to the tree \(T\), \(\gamma(T – e) = \gamma(T)\). We present a constructive characterization of \({E}_0\).

A procedure based on the Kronecker product yields \(\pm 1\)-matrices \(X,Y\) of order \(8mn\), satisfying
\(XX^t + YY^t = 8mnI \quad {and} \quad XY^t = YX^t = 0,\)
given Hadamard matrices of orders \(4m\) and \(4n\). This allows the construction of some infinite classes of Hadamard matrices – and in particular orders \(8mnp\), for values of \(p\) including (for \(j \geq 0\)) \(5, 9^j, 25, 9^j, \), improving the usual Kronecker product construction by at least a factor of \(2\). A related construction gives Hadamard matrices in orders \(4 \cdot 5^i \cdot 9^j, 0 \leq i \leq 4\). To this end we introduce some disjoint weighing matrices and exploit certain Williamson matrices studied by Turyn and Xia. Some new constructions are given for symmetric and skew weighing matrices, resolving the case of skew \(W(N, 16)\) for \(N = 30, 34, 38\).

The set of Lyndon words of length \(N\) is obtained by choosing those strings of length \(n\) over a finite alphabet which are lexicographically least in the aperiodic equivalence classes determined by cyclic permutation. We prove that interleaving two Lyndon words of length \(n\) produces a Lyndon word of length \(2n\). For the binary alphabet \(\{0, 1\}\) we represent the set of Lyndon words of length \(n\) as vertices of the \(n\)-cube. It is known that the set of Lyndon words of length \(n\) form a connected subset of the \(n\)-cube. A path of vertices in the \(n\)-cube is a list of strings of length \(n\) in which adjacent strings differ in a single bit. Using paths of Lyndon words in the \(n\)-cube we construct longer paths of Lyndon words in higher order cubes by shuffling and concatenation.

A tricover of pairs by quintuples of a \(v\)-set \(V\) is a family of \(5\)-subsets of \(V\) (called blocks) with the property that every pair of distinct elements from \(V\) occurs in at least three blocks. If no other such tricover has fewer blocks, the tricover is said to be minimum, and the number of blocks in a minimum tricover is the covering number \(C_3(v, 5, 2)\), or simply \(C_3(v)\). It is well known that\(C_3(v) \geq \lceil \frac{{v} \lceil \frac {3(v-1)}{4} \rceil} {5} \rceil = B_3(v)\) , where \(\lceil x \rceil\) is the least integer not less than \(x\). It is shown here that if \(v \equiv 0 \pmod{4}\) and \(v \geq 8\), then \(C_3(v) = B_3(v)\).

The concept of rectangular designs with varying replicates is introduced. A class of such designs is constructed with an example.

We study the isomorphic factorization of complete bipartite graphs into trees. It is known that for complete bipartite graphs, the divisibility condition is also a sufficient condition for the existence of isomorphic factorization. We give necessary and sufficient conditions for the divisibility, that is, necessary and sufficient conditions for a pair \([m,n]\) such that \(mn\) is divisible by \((m+n-1)\), and investigate structures of the set of pairs \([m,n]\) satisfying divisibility. Then we prove that the divisibility condition is also sufficient for the existence of an isomorphic tree factor of a complete bipartite graph by constructing the tree dividing \(K({m,n})\).