Groups provide the mathematical language for exact symmetry. Applications in biology and other fields are now raising the problem of developing a rigorous theory of approximate symmetry. In this paper, it is shown how approximate symmetry is determined by a quasigroup.

A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices \(x\), the subgraph induced by the neighbour set of \(x\) is isomorphic to the subgraph induced by the closed non-neighbour set of \(x\). Graphs with the ncc property were characterized in [1] by the existence of a locally \(C_4\) perfect matching \(M\): every two edges of \(M\) induce a subgraph isomorphic to \(C_4\). In the present article, we investigate variants of locally \(C_4\) perfect matchings. We consider the cases where pairs of distinct edges of the matching induce isomorphism types including \(P_4\), the paw, or the diamond. We give several characterizations of graphs with such matchings. In addition, we supply characterizations of graphs with matchings whose edges satisfy a prescribed parity condition.

In this paper we obtain some necessary conditions for the existence of balanced arrays (B-arrays) with two symbols and having strength seven. We then describe how these conditions involving the parameters of the array can be used to obtain an upper bound on the constraints of such arrays, and give some illustrative examples to this effect.

A defensive alliance in a graph \( G(V,E) \) is a set of vertices \( S \subseteq V \) such that for every vertex \( v \in S \), the closed neighborhood \( N_G[v] \) of \( v \) has at least as many vertices in \( S \) as it has in \( V – S \). An offensive alliance is a set of vertices \( S \subseteq V \), such that for every vertex \( v \) in the boundary \( \partial(S) \) of \( S \) the number of neighbors that \( v \) has in \( S \) is greater than or equal to the number of neighbors it has in \( V – S \). A subset of vertices which is both an offensive and a defensive alliance is called a powerful alliance. An alliance which is also a dominating set is called a global alliance. In this paper, we show that finding an optimal defensive (offensive, powerful) global alliance is an NP-hard problem.

We discuss a branch of Ramsey theory concerning vertex Folkman numbers and how computer algorithms have been used to compute a new Folkman number. We write \( G \rightarrow (a_1, \ldots, a_k)^v \) if for every vertex \( k \)-coloring of an undirected simple graph \( G \), a monochromatic \( K_{a_i} \) is forced in color \( i \in \{1, \ldots, k\} \). The vertex Folkman number is defined as\(F_v(a_1, \ldots, a_k; p) = \text{min}\{|V(G)| : G \rightarrow (a_1, \ldots, a_k)^v \wedge K_p \nsubseteq G\}.\) Folkman showed in 1970 that this number exists for \( p > \text{max}\{a_1, \ldots, a_k\} \). Let \( m = 1 + \sum_{i=1}^k (a_i – 1) \) and \( a = \text{max}\{a_1, \ldots, a_k\} \), then \(F_v(a_1, \ldots, a_k; p) = m \text{ for } p > m,\) and \(F_v(a_1, \ldots, a_k; p) = a + m \text{ for } p = m.\)For \( p < m \) the situation is more difficult and much less is known. We show here that, for a case of \( p = m – 1 \), \( F_v(2, 2, 3; 4) = 14 \).

By analogy with Stirling numbers, tri-restricted numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most three elements. Tri-restricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of tri-restricted numbers of the second kind. A new recurrence relation for the tri-restricted numbers of the second kind is presented, with a combinatorial proof. In answer to a problem that has remained open for several years, it is then shown by determinantal techniques that the tri-restricted numbers of the first kind also satisfy a recurrence relation. This relation is used to obtain a reciprocity theorem connecting the two kinds of tri-restricted number.