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 \(\rho_1(D) = c\).

A graph of even order is called path-pairable if, for any pairing of its vertices, there exist edge-disjoint paths connecting the paired vertices. Extremal problems for path-pairable graphs with restrictions on the maximum degree will be considered. In particular, let \(f(n, k)\) denote the minimum number of edges in a path-pairable graph of order \(n\) and maximum degree \(k\). Exact values of \(f(n, k)\) are determined for \(k = n-1, n-2,\) and \(n-3\).

Using the characterization of those prime powers \(q\) for which \({GF}(q)\) admits a quadratic starter: i.e. a pairing \((x_i, y_i)\), \(i = 1, 2, \ldots, \frac{q-1}{2}\), of nonzero squares \(x_i\) with non-squares \(y_i\) in \({GF}(q)\) such that the differences \(\pm(x_i – y_i)\) are all distinct, we obtain a new infinite family of nested row-column designs.

Zigzag functions were defined by Brassard, Crépeau, and Sántha [1] in connection with an application to the construction of oblivious transfers (a useful tool in cryptographic protocols). They proved that linear zigzag functions are equivalent to self-intersecting codes, which have been studied by several researchers.In this paper, we begin an investigation of general (linear or nonlinear) zigzag functions.In particular, we prove some bounds (i.e., necessary conditions for the existence of zigzag functions) that generalize known bounds for linear zigzag functions.

In the last two decades, mathematicians have discussed various transivities of automorphism groups of designs (i.e., point, block, and flag transivities), from all these studies, we know that
\[
0 \leq O^{\#}(G, \mathbf{B}) – O^{\#}(G, \mathbf{X}) \leq |\mathbf{B}| – |\mathbf{X}|
\]
for \(2-(v, k, \lambda)\) designs (see \([\)BMP\]). In this paper, we discuss the orbit structure of general combinatorial designs \(\mathbf{D}(\mathbf{X}, \mathbf{B})\) and obtain the equalities \[O^{\#}(G, \mathbf{F}) = \sum\limits_{i=1}^{u} O^{\#}(H(x_i), X_{i}) =\sum\limits_{j=1}^{l} O^{\#}(H(B_j), B_j),
\]
where \(H(x_i)\) and \(H(B_j)\) are the stabilizers of the point \(x_i\) and the block \(B_j\) respectively, \(u = O^{\#}(G, \mathbf{X})\), \(l = O^{\#}(G, \mathbf{B})\).

The problem of determining which graphs have the property that every maximal independent set of vertices is also a maximum independent set was proposed by M.D. Plummer
in 1970 [28]. This was partly motivated by the observation that whereas determining the independence number of an arbitrary graph is NP-complete, for a well-covered graph one can
simply apply the greedy algorithm. Although a good deal of effort has been expended in an
attempt to obtain a complete characterization of such graphs, that result appears as elusive as ever. In this paper, intended to serve as an introduction to the problem, several of the main attacks will be highlighted with particular emphasis on the approach involving the girth of such graphs.

We consider whether an order-ten Latin square with an order-four Latin subsquare can belong to an orthogonal triple of Latin squares. We eliminate \(20\) of \(28\) possibilities for how this could occur by considering the structure of possible mates. Our technique supplements the small collection of existing tools for obtaining negative results regarding
the existence of collections of orthogonal Latin squares.

The partitions into Baer subplanes of the Desarguesian projective planes of order \(9\), \(16\), and \(25\) are classified by computer. It is also shown that the non-Desarguesian projective planes of order \(9\) and the non-Desarguesian translation planes of order \(16\) and \(25\) do not admit such a partition.