We determine those triples \((g, u, k)\) for which there exists a pair of group divisible designs with block size \(3\), each on the same \(u\) groups of size \(g\), having exactly \(k\) blocks in common.

Using the explicit determination of all ovals in the 3 non-Desarguesian projective planes of order 9 given in [4] or [8], we prove that there are no other Benz planes of order 9 than the three miquelian planes and the Minkowski plane over the Dickson near-field of type \(\{3,2\}\).

Sufficient conditions depending on the minimum degree and the independence number of a simple graph for the existence of a \(k\)-factor are established.

In this paper, we shall establish some construction methods for resolvable Mendelsohn designs, including constructions of the product type. As an application,we show that the necessary condition for the existence of a \((v, 4, \lambda)\)-RPMD, namely,
\(v \equiv 0\) or \(1\) (mod 4), is also sufficient for \(\lambda > 1\) with the exception of pairs \((v,\lambda)\)
where \(v = 4\) and \(\lambda\) odd. We also obtain a (v, 4, 1)-RPMD for \(v = 57\) and \(93\).

A counterexample is presented to the following conjecture of Jackson: If \(G\) is a 2-connected graph on at most \(3k + 2\) vertices with degree sequence \((k, k, \ldots, k, k+1, k+1)\), then \(G\) is hamiltonian.

We provide graceful and harmonious labelings for all vertex deleted and edge-deleted prisms. We also show that with the sole exception of the cube all prisms are harmonious.

Let \(G\) be a 2-connected simple graph of order \(n\) (\(\geq 3\)) with connectivity \(k\). One of our results is that if there exists an integer \(t\) such that for any distinct vertices \(u\) and \(v\), \(d(u,v) = 2\) implies \(|N(u) \bigcup N(v)| \geq n-t\), and for any independent set \(S\) of cardinality \(k+1\), \(\max\{d(u) \mid u \in S\} \geq t\), then \(G\) is hamiltonian. This unifies many known results for hamiltonian graphs. We also obtain a similar result for hamiltonian-connected graphs.

A graph \(G = (V(G), E(G))\) is the competition graph of an acyclic digraph \(D = (V(D), A(D))\) if \(V(G) = V(D)\) and there is an edge in \(G\) between vertices \(x, y \in V(G)\) if and only if there is some \(v \in V(D)\) such that \(xv, yv \in A(D)\). The competition number \(k(G)\) of a graph \(G\) is the minimum number of isolated vertices needed to add to \(G\) to obtain a competition graph of an acyclic digraph. Opsut conjectured in 1982 that if \(\theta(N(v)) \leq 2\) for all \(v \in V(G)\), then the competition number \(k(G)\) of \(G\) is at most \(2\), with equality if and only if \(\theta(N(v)) = 2\) for all \(v \in V(G)\). (Here, \(\theta(H)\) is the smallest number of cliques covering the vertices of \(H\).) Though Opsut (1982) proved that the conjecture is true for line graphs and recently Kim and Roberts (1989) proved a variant of it, the original conjecture is still open. In this paper, we introduce a class of so-called critical graphs. We reduce the question of settling Opsut’s conjecture to the study of critical graphs by proving that Opsut’s conjecture is true for all graphs which are disjoint unions of connected non-critical graphs. All \(K_4\)-free critical graphs are characterized. It is proved that Opsut’s conjecture is true for critical graphs which are \(K_4\)-free or are \(K_4\)-free after contracting vertices of the same closed neighborhood. Some structural properties of critical graphs are discussed.

We investigate the existence of \(a\)-valuations and sequential labelings for a variety of grids in the plane, on a cylinder and on a torus.

Let \(G\) be a simple graph of order \(n\) with independence number \(\alpha\). We prove in this paper that if, for any pair of nonadjacent vertices \(u\) and \(v\), \(d(u)+d(v) \geq n+1\) or \(|N(u) \cap N(v)| \geq \alpha\), then \(G\) is \((4, n-1)\)-connected unless \(G\) is some special graphs. As a corollary, we investigate edge-pancyclicity of graphs.