Using computer algorithms, we show that in any \(2-(22, 8, 4)\) design, there are no blocks of type \(3\), thus leaving as possible only types \(1\) and \(2\).
Blocks of type \(3\), i.e., those which intersect two others in one point, are eliminated using the algorithms described in our previous paper. It was perhaps the second largest computation ever performed (after the solution to the RSA-129 challenge), requiring more than a century of cpu time.

A transitive orientation of a partial triple system \((S,T)\) of index \(2\lambda\) is a partial transitive triple system formed by replacing each triple \(t \in T\) with a transitive triple defined on the same vertex set as \(t\), such that each ordered pair occurs in at most \(\lambda\) of the resulting transitive triples. A transitive orientation \((S_1,T_1)\) of \((S,T)\) is said to be balanced if for all \(\{u,v\} \subseteq S\), if \(\{u,v\}\) occurs in \(\ell\) triples in \(T\) then \(\left\lfloor{\ell}/{2}\right\rfloor\)
and \(\left\lceil{\ell}/{2}\right\rceil\) transitive triples in \(T_1\) contain the arcs \((u,v)\) and \((v,u)\) respectively. In this paper, it is shown that every partial triple
system has a balanced transitive orientation. This result is then used to prove the existence of certain transitive group divisible designs.

The Prime Power Conjecture (PPC) states that abelian planar difference sets of order \(n\) exist only for \(n\) a prime power. Lander et al. have shown that orders divisible by certain composites can be eliminated. In this paper, we show how to extend this list of excluded orders.

A critical set \(C\) in a Latin square \(L\) is a partial Latin square which has a unique completion to \(L\) and for which no subset of \(C\) has this property. In this paper, I document known results on the possible sizes of critical sets, and provide a reference list for the existence of critical sets in Latin squares of order less than or equal to \(10\).Many of the results in this list are new, and where this is the case, I exhibit a critical set of the given size in the Appendix.

Let \(S^n\) be the \(n\)-dimensional sphere and \(K\) be the simplicial complex consisting of all faces of some \((n+1)\)-dimensional simplex. We present an explicit construction of a function \(g: S^n \to |K|\) such that for every \(x \in S^n\), the supports of \(g(x)\) and \(g(-x)\) are disjoint. This construction provides a new proof of the following result of Bajméczy and Bérdny \([1]\), which is a generalization of a theorem of Radon \([4]\):If \(f: |K| \to \mathbb{R}^n\) is a continuous map, then there are two disjoint faces \(\Delta_1, \Delta_2\) of \(\Delta\) such that \(f(\Delta_1) \cap f(\Delta_2) \neq \emptyset\).

It is well known that one can construct a family of \(\frac{q^2 – q}{2}\) Miquelian inversive planes on the same point set such that any two share exactly the blocks through a fixed point. Further, Ebert [10] has shown that this family can be augmented for even \(q\) by adding some Suzuki-Tits inversive planes. We wish to apply the method of Ebert combined with a technique from Dover [6] to obtain a family of unitals which have the same property.

In this paper, we first generalize a classical result of B. Toft (\(1974\)) on \(r\)-type-constructions for graphs (rather than hypergraphs). We then demonstrate how this result can be utilized to construct colour-critical graphs, with a special focus on
\(\Delta\)-colour-critical graphs. This generalization encompasses most known constructions that generate small critical graphs. We also obtain upper bounds for the minimum excess function \(\eta(k,p)\) when \(4 \leq k \leq 6\); where
\[
\eta(k,p) = \min\limits_{G\in K(k,p)} \epsilon(G),
\]
in which \(\epsilon(G) = 2|E(G)| – |V(G)|(k-1)\), and \(K(k,p)\) is the class of all
\(k\)-colour-critical graphs on \(p\) vertices with \(\Delta = k\). Using these techniques, we construct an infinite family of \(\Delta\)-colour-critical graphs for \(\Delta = 5\) with a relatively small minimum excess function; Furthermore, we prove that \(\eta(6, 6n) \leq 6(n-1)\) (\(n\geq2\)) which shows that there exists an infinite family of \(\Delta\)-colour-critical graphs for \(\Delta = 6\).

An open dominating set for a digraph \(D\) is a subset \(S\) of vertices of \(D\) such that every vertex of \(D\) is adjacent from some vertex of \(S\). The cardinality of a minimum open dominating set for \(D\) is the open domination number \(\rho_1(D)\) of \(D\). The lower orientable open domination number \(\text{dom}_1(G)\) of a graph \(G\) is the minimum open domination number among all orientations of \(G\). Similarly, the upper orientable open domination number \(\text{DOM}_1(G)\) of \(G\) is the maximum such open
domination number. For a connected graph \(G\), it is shown that \(\text{dom}_1(G)\) and \(\text{DOM}_1(G)\) exist if and only if \(G\) is not a tree. A discussion of the upper orientable open domination number of complete graphs is given. It is shown that for each integer \(c\) with \(\text{dom}_1(K_n) \leq c \leq \text{DOM}_1(K_n)\), there exists an
orientation \(D\) of \(K_n\) such that \(\rho_1(D) = c\).