We prove that the domination number of every graph of diameter 2 on \(n\) vertices is at most \(\left(\frac{1}{\sqrt{2}} + o(1)\right) \sqrt{n \log n}\) as \(n \to \infty\) (with logarithm of base \(e\)). This result is applied to prove that if a graph of order \(n\) has diameter 2, then it contains a spanning caterpillar whose diameter does not exceed \(\left(\frac{3}{\sqrt{2}} + o(1)\right) \sqrt{n \log n}\). These estimates are tight apart from a multiplicative constant, since there exist graphs of order \(n\) and diameter 2, with domination number not smaller than \(\left(\frac{1}{2\sqrt{2}} + o(1)\right) \sqrt{n \log n}\). In contrast, in graphs of diameter 3, the domination number can be as large as \(\lfloor \frac{n}{2} \rfloor\) (but not larger).

Our results concerning diameter 2 improve the previous upper bound of \(O(n^{3/4})\), published by Faudree et al. in [Discuss. Math. Graph Theory 15 (1995), 111-118].

As an extension of the fractional domination and fractional domatic graphical parameters, multi-fractional domination parameters are introduced. We demonstrate the Linear Programming formulations, and to these formulations we apply the Partition Class Theorem, which is a generalization of the Automorphism Class Theorem. We investigate some properties of the multi-fractional domination numbers and their relationships to the fractional domination and fractional domatic numbers.

In a graph, the Steiner distance of a set of vertices \(U\) is the minimum number of edges in a connected subgraph containing \(U\). For \(k \geq 2\) and \(d \geq k-1\), let \(S(k,d)\) denote the property that for all sets \(S\) of \(k\) vertices with Steiner distance \(d\), the Steiner distance of \(S\) is preserved in any induced connected subgraph containing \(S\). A \(k\)-Steiner-distance-hereditary (\(k\)-SDH) graph is one with the property \(S(k, d)\) for all \(d\). We show that property \(S(k, k)\) is equivalent to being \(k\)-SDH, and that being \(k\)-SDH implies \((k + 1)\)-SDH. This establishes a conjecture of Day, Oellermann and Swart.

The quantity \(g^{(k)}(v)\) was introduced in [4] as the minimum number of blocks necessary in a pairwise balanced design on \(v\) elements, subject to the condition that the longest block have cardinality \(k\). When \(k \geq (v – 1)/2\), it is known that \(g^{(k)}(v) = 1 + (v – k)(3k – v + 1)/2\), except for the case when \(v \equiv 1 \pmod{4}\) and \(k = (v – 1)/2\). This exceptional “case of first failure” was treated in [1] and [2]. In this paper, we discuss the structure of the “case of first failure” for the situation when \(v = 4s + 4\).

We construct some codes, designs and graphs that have the first or second Janko group, \(J_1\) or \(J_2\), respectively, acting as an automorphism group. We show computationally that the full automorphism group of the design or graph in each case is \(J_1\), \(J_2\) or \(\bar{J}_2\), the extension of \(J_2\) by its outer automorphism, and we show that for some of the codes the same is true.

A 3-regular graph \(G\) is called a 3-circulant if its adjacency matrix \(A(G)\) is a circulant matrix. We show how all disconnected 3-circulants are made up of connected 3-circulants and classify all connected 3-circulants as one of two basic types. The rank of \(A(G)\) is then completely determined for all 3-circulant graphs \(G\).

The independence number \(\beta_n\), for knights on equilateral triangular boards \(T_n\), of regular hexagons is determined for all \(n\).

It was conjectured by Lee that a cubic simple graph with \(4k + 2\) vertices is edge-magic [5]. In this paper we show that the conjecture is not true for multigraphs or disconnected simple graphs in general. Several new classes of cubic edge-magic graphs are exhibited.

In 1976 Erdős asked about the existence of Steiner triple systems that lack collections of \(j\) blocks employing just \(j+2\) points. This has led to the study of anti-Pasch, anti-mitre and 5-sparse Steiner triple systems. Simultaneously generating sets and bases for Steiner triple systems and \(t\)-designs have been determined. Combining these ideas, together with the observation that a regular graph is a 1-design, we arrive at a natural definition for the girth of a design. In turn, this provides a natural extension of the search for cages to the universe of all \(t\)-designs. We include the results of computational experiments that give an abundance of examples of these new definitions.

A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(x, y,\) and \(z\) with \(d(x,y) = 2\) and \(z \in N(x) \cap N(y)\), \(d(x) + d(y) \geq |N(x) \cup N(y) \cup N(z)| – 1\). Let \(G\) be a \(3\)-connected \(L_1\)-graph of order \(n \geq 18\). If \(\delta(G) \geq n/3\), then every pair of vertices \(u\) and \(v\) in \(G\) with \(d(u,v) \geq 3\) is connected by a Hamiltonian path of \(G\).