We provide upper estimates on the weak exponent of indecomposability of an irreducible Boolean matrix.

The toughness \(t(G)\) of a noncomplete graph \(G\) is defined as

\[t(G) = \min\left\{\frac{|S|}{\omega(G-S)} \mid S \subseteq V(G), \omega(G-S) \geq 2\right\},\]

where \(\omega(G-S)\) is the number of components of \(G-S\). We also define \(t(K_n) = +\infty\) for every \(n\).

The middle graph \(M(G)\) of a graph \(G\) is the graph obtained from \(G\) by inserting a new vertex into every edge of \(G\) and by joining by edges those pairs of these new vertices which lie on adjacent edges of \(G\).

In this article, we give the toughness of the middle graph of a graph, and using this result we also give a sufficient condition for the middle graph to have a \(k\)-factor.

This paper gives constructions of balanced incomplete block designs and group divisible designs with \(k = 7, 8,\) or \(9\), and \(\lambda = 1\). The first objective is to give constructions for all possible cases with the exception of \(40, 78,\) and \(157\) values of \(v\). Many of these initial exceptions have now been removed by Abel. In an update section, more are removed; group divisible designs with groups of size \(k(k-1)\) are constructed for \(k = 7\) and \(8\) with \(124\) and \(87\) exceptions; it is also established that \(v \geq 294469\) and \(v \equiv 7\) mod \(42\) suffices for the existence of a resolvable balanced incomplete block design with \(k = 7\). Group divisible designs with group size \(k\) and resolvable designs are constructed.

In this paper, we obtain critical sets for the general dihedral group, but we are not able to decide whether they are minimal. We also show the existence of a weakly completable critical set in the latin square based on the dihedral group of order six. We believe this to be the smallest group-based square to have such a set.

An \(S_h\)-set (mod \(m\)) is a set \(S\) of integers such that the sums\(a_1 + a_2 + \cdots + a_h\) of elements \(a_1 \leq a_2 \leq \cdots 1\) and prove that equality is possible at least when \(h=p\) is a prime (Theorem).

We investigate those classes \(\mathcal{K}\) of relational structures closed under operations that are defined by excluding a fixed class of finite structures. We characterize such classes and show they contain an infinite family of pairwise non-embeddable members. NEC structures are defined by certain extension conditions. We construct countable universal structures in \(\mathcal{K}\) satisfying only finitely many of the NEC extension conditions.

The notion of normal quotient of a vertex-transitive graph was introduced in [5]. It was shown there that many graph properties are inherited by normal quotients. The definition of a normal quotient was given in [5] in group-theoretical terms. In this note we give a combinatorial approximation to this notion which extends the original definition. We show that many of the properties that were inherited by group-theoretical normal quotients are also inherited by combinatorial ones.

A \((k;g)\)-cage is a smallest \(k\)-regular graph with girth \(g\). Harary and Kovacs [2] conjectured that for all \(k \geq 3\) and odd \(g \geq 5\), there exists a \((k;g)\)-cage which contains a cycle of length \(g+1\). Among other results, we prove the conjecture for all \(k \geq 3\) and \(g \in \{5,7\}\).

The toughness \(t(G)\) of a noncomplete graph \(G\) is defined as

\[t(G) = \min{\left\{\frac{|S|}{\omega(G-S)} \mid S \subset V(G), \omega(G-S) \geq 2\right\}}\]

where \(\omega(G-S)\) is the number of components of \(G-S\). We also define \(t(K_n) = +\infty\) for every \(n\).

In this article, we discuss the toughness of the endline graph of a graph and the middle graph of a graph.

We present several new non-isomorphic one-factorizations of \(K_{36}\) and \(K_{40}\) which were found through hill-climbing and testing Skolem sequences. We also give a brief comparison of the effectiveness of hill-climbing versus exhaustive search for perfect one-factorizations of \(K_{2n}\) for small values of \(2n\).