Quackenbush [5] has studied the properties of squags or “Steiner quasigroups”, that is, the corresponding algebra of Steiner triple systems. He has proved that if a finite squag \((P; \cdot)\) contains two disjoint subsquags \((P_1; \cdot)\) and \((P_2; \cdot)\) with cardinality \(|P_1| = |P_2| = \frac{1}{3} |P|\), then the complement \(P_3 = P – (P_1 \cup P_2)\) is also a subsquag and the three subsquags \(P_1, P_2\) and \(P_3\) are normal. Quackenbush then asks for an example of a finite squag of cardinality \(3n\) with a subsquag of cardinality \(n\), but not normal. In this paper, we construct an example of a squag of cardinality \(3n\) with a subsquag of cardinality \(n\), but it is not normal; for any positive integer \(n \geq 7\) and \(n \equiv 1\) or \(3\) (mod \(6\)).

A plane graph is an embedding of a planar graph into the sphere which may have multiple edges and loops. A face of a plane graph is said to be a pseudo triangle if either the boundary of it has three distinct edges or the boundary of it consists of a loop and a pendant edge. A plane pseudo triangulation is a connected plane graph of which each face is a pseudo triangle. If a plane pseudo triangulation has neither a multiple edge nor a loop, then it is a plane triangulation. As a generalization of the diagonal flip of a plane triangulation, the diagonal flip of a plane pseudo triangulation is naturally defined. In this paper we show that any two plane pseudo triangulations of order \(n\) can be transformed into each other, up to ambient isotopy, by at most \(14n – 64\) diagonal flips if \(n \geq 7\). We also show that for a positive integer \(n \geq 5\), there are two plane pseudo triangulations with \(n\) vertices such that at least \(4n – 15\) diagonal flips are needed to transform into each other.

An extended Mendelsohn triple system of order \(v\) with a idempotent element (EMTS(\(v, a\))) is a collection of cyclically ordered triples of the type \(\{x, y, z\}\), \(\{x, x, y\}\) or \(\{x, x, x\}\) chosen from a \(v\)-set, such that every ordered pair (not necessarily distinct) belongs to only one triple and there are \(a\) triples of the type \(\{x, x, x\}\). If such a design with parameters \(v\) and \(a\) exist, then they will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). A necessary and sufficient condition for the existence of EMTS(\(v, 0\)) and EMTS(\(v, 1\)) are \(v \equiv 0\) (mod \(3\)) and \(v \not\equiv 0\) (mod \(3\)), respectively. In this paper, we have constructed two EMTS(\(v, 0\))’s such that the number of common triples is in the set \(\{0, 1, 2, \ldots, b_{v, 0} – 3, b_{v, 0}\}\), for \(v \equiv 0\) (mod \(3\)). Secondly, we have constructed two EMTS(\(v, 1\))’s such that the number of common triples is in the set \(\{0, 1, 2, \ldots, b_{v, 1} – 2, b_{v, 1}\}\), for \(v \not\equiv 0\) (mod \(3\)).

A Latin square is \(N_e\) if it has no intercalates (Latin subsquares of order \(2\)). We correct results published in an earlier paper by McLeish, dealing with a construction for \(N_2\) Latin squares.

In [13], we conjectured that if \(G = (V_1, V_2; E)\) is a bipartite graph with \(|V_1| = |V_2| = 2k\) and minimum degree at least \(k + 1\), then \(G\) contains \(k\) vertex-disjoint quadrilaterals. In this paper, we propose a more general conjecture: If \(G = (V_1, V_2; E)\) is a bipartite graph such that \(|V_1| = |V_2| = n \geq 2\) and \(\delta(G) \geq [n/2] + 1\), then for any bipartite graph \(H = (U_1, U_2; F)\) with \(|U_1| \leq n, |U_2| \leq n\) and \(\Delta(H) \leq 2, G\) contains a subgraph isomorphic to \(H\). To support this conjecture, we prove that if \(n = 2k + t\) with \(k \geq 0\) and \(t \geq 3, G\) contains \(k + 1\) vertex-disjoint cycles covering all the vertices of \(G\) such that \(k\) of them are quadrilaterals.

In a finite projective plane, a \(k\)-arc \(\mathcal{K}\) covers a line \(l_0\) if every point on \(l_0\) lies on a secant of \(\mathcal{K}\). Such \(k\)-arcs arise from determining sets of elements for which no linear \((n, q, t)\)-perfect hash families exist [1], as well as from finding sets of points in \(\mathrm{AG}(2, q)\) which determine all directions [2]. This paper provides a lower bound on \(k\) and establishes exactly when the lower bound is attained. This paper also gives constructions of such \(k\)-arcs with \(k\) close to the lower bound.

In this paper we determine the \(k\)-domination number \(\gamma_k\) of \(P_{2k+2} \times P_n\) and \(\lim_{{m,n} \to \infty} \frac{\Gamma_k(P_m \times P_n)}{mn}\).

A digraph obtained by replacing each edge of a complete \(n\)-partite graph by an arc or a pair of mutually opposite arcs is called a semi-complete \(n\)-partite digraph. An \(n\)-partite tournament is an orientation of a complete \(n\)-partite graph. In this paper we shall prove that a strongly connected semicomplete \(n\)-partite digraph with a longest directed cycle \(C\), contains a spanning strongly connected \(n\)-partite tournament which also has the longest directed cycle \(C\) with exception of a well determined family of semicomplete bipartite digraphs. This theorem shows that many well-known results on strongly connected \(n\)-partite tournaments are also valid for strongly connected semicomplete \(n\)-partite digraphs.