In this paper, we show that group divisible designs with block size five, group-type and index odd exist with a few possible exceptions.

A digraph \(D\) is called semicomplete \(c\)-partite if its vertex set \(V(D)\) can be partitioned into \(c\) sets (partite sets) such that for any two vertices \(x\) and \(y\) in different partite sets, at least one arc between \(x\) and \(y\) is in \(D\) and there are no arcs between vertices in the same partite set. The path covering number of \(D\) is the minimum number of paths in \(D\) that are pairwise vertex disjoint and cover the vertices of \(D\). Volkmann (1996) has proved two sufficient conditions on hamiltonian paths in semicomplete multipartite digraphs and conjectured two related sufficient conditions. In this paper, we derive sufficient conditions for a semicomplete multipartite digraph to have path covering number at most \(k\) and show that Volkmann’s results and conjectures can be readily obtained from our conditions.

The Fibonacci number of a graph is the number of independent sets of the graph. In this paper, we compute algorithmically the Fibonacci numbers of lattice product graphs.

In this note, we solve a conjecture of Dénes, Mullen, and Suchower [2] on power sets of Latin squares.

In this article, we construct a large set of idempotent quasigroups of order 62. The spectrum for large sets of idempotent quasigroups of order \(n\) (briefly, \(LQ(n)\)) is the set of all integers \(n \geq 3\) with the exception \(n = 6\) and the possible exception \(n = 14\).

We settle the existence status of some previously open cases of abelian difference sets. Our results fill ten missing entries in the recent table of Lepez and Sanchez, all with answer `No’.

Recently, Raines and Rodger have proved that for all \(\lambda \geq 1\), any partial extended triple system of order \(n\) and index \(\lambda\) can be embedded in a (complete) extended triple system of order \(v\) and index \(\lambda\) for any even \(v \geq 4n + 6\). In this note, it is shown that if \(\lambda\) is even then this bound on \(v\) can be improved to all \(v \geq 3n + 5\), and under some conditions to all \(v \geq 2n + 1\).

It is shown that if a graph \(G\) is connected, claw-free, and such that the vertices of degree 1 of every induced bull have a common neighbor in \(G\), then \(G\) is traceable.

Some extremal set problems can be phrased as follows. Given an \(m \times n\) \((0,1)\)-matrix \(A\) with no repeated columns and with no submatrix of a certain type, what is a bound on \(n\) in terms of \(m\)? We examine a conjecture of Frankl, Füredi, and Pach and the author that when we forbid a \(k \times l\) submatrix \(F\) then \(n\) is \(O(m^{k})\). Two proof techniques are presented, one is amortized complexity and the other uses a result of Alon to show that \(n\) is \(O(m^{2k-1-\epsilon})\) for \(\epsilon=(k-1)/(13 \log_2 l)\), improving on the previous bound of \(O(m^{2k-1})\).

A graph \(H\) is \(G\)-decomposable if \(H\) can be decomposed into subgraphs, each of which is isomorphic to \(G\). A graph \(G\) is a greatest common divisor of two graphs \(G_1\) and \(G_2\) if \(G \) is a graph of maximum size such that both \(G_1\) and \(G_2\) are \(G\)-decomposable. The greatest common divisor index of a graph \(G\) of size \(q\) is the greatest positive integer \(n\) for which there exist graphs \(G_1\) and \(G_2\), both of size at least \(nq\), such that \(G\) is the unique greatest common divisor of \(G_1\) and \(G_2\). The corresponding concepts are defined for digraphs. Relationships between greatest common divisor index for a digraph and for its underlying graph are studied. Several digraphs are shown to have infinite index, including matchings, short paths, union of stars, transitive tournaments, the oriented 4-cycle. It is shown that for \(5 \leq p \leq 10\), if a graph \(F\) of sufficiently large size is \(C_p\)-decomposable, then \(F\) is also \((P_{p-1} \cup P_3)\)-decomposable. From this it follows that the even cycles \(C_6\), \(C_8\) and \(C_{10}\) have finite greatest common divisor index.