In this paper, we prove that a \(V(3, t)\) exists for any prime power \(3t + 1\), except when \(t = 5\), as no \(V(3, 5)\) exists.

In this paper, we survey some recent bounds on domination parameters. A characterisation of connected graphs with minimum degree at least 2 and domination number exceeding a third their size is obtained. Upper bounds on the total domination number, \(\gamma_t(G)\), of a graph \(G\) in terms of its order and size are established. If \(G\) is a connected graph of order \(n\) with minimum degree at least 2, then either \(\gamma_t(G) \leq 4n/7\) or \(G \in \{C_3,C_5,C_6,C_{10}\}\). A characterisation of those graphs of order \(n\) which are edge-minimal with respect to satisfying \(G\) connected, \(\delta(G) \geq 2\), and \(\gamma(G) \geq 4n/7\) is obtained. We establish that if \(G\) is a connected graph of size \(q\) with minimum degree at least 2, then \(\gamma_t(G) \leq (q + 2)/2\). Connected graphs \(G\) of size \(q\) with minimum degree at least 2 satisfying \(\gamma_t(G) > q/2\) are characterised. Upper bounds on other domination parameters, including the strong domination number and the restrained domination number are presented. We provide a constructive characterisation of those trees with equal domination and restrained domination numbers. A constructive characterisation of those trees with equal domination and weak domination numbers is also obtained.

Necessary and sufficient conditions for the existence of a decomposition of \(\lambda K_v\) into edge-disjoint copies of the Petersen graph are proved.

A \((v,k,t)\) trade \(T = T_1 – T_2\) of volume \(m\) consists of two disjoint collections \(T_1\) and \(T_2\), each containing \(m\) blocks (\(k\)-subsets) such that every \(t\)-subset is contained in the same number of blocks in \(T_1\) and \(T_2\). If each \(t\)-subset occurs at most once in \(T_1\), then \(T\) is called a Steiner \((k,t)\) trade. In this paper, the spectrum (that is, the set of allowable volumes) of Steiner trades is discussed, with particular reference to the case \(t = 2\). It is shown that the volume of a Steiner \((k, 2)\) trade is at least \(2k – 2\) and cannot equal \(2k – 1\). We show how to construct a Steiner \((k, 2)\) trade of volume \(m\) when \(m \geq 3k – 3\), or \(m\) is even and \(2k – 2 \leq m \leq 3k – 4\). For \(k = 5\) or \(6\), the non-existence of Steiner \((k,2)\) trades of volume \(2k + 1\) is demonstrated, and for \(k = 7\), we exhibit a Steiner \((k,2)\) trade of volume \(2k + 1\). In addition, the structure of Steiner \((k,2)\) trades of volumes \(2k – 2\) and \(2k\) (\(k \neq 3,4\)) is shown to be unique. A generalisation of our constructions to trades with blocks based on arbitrary simple graphs is also presented.

This paper characterizes a particular scheme of partially filled Latin squares and when they can be completed to full Latin squares. In particular, given an \(n \times n\) array with the first \(s\) rows and the first \(d\) cells of row \(s+1\) filled with \(n\) distinct symbols in such a way that no symbol occurs more than once in any row or column, necessary and sufficient conditions are found for when this array can be completed to a full Latin square.

We give counterexamples for two theorems given for the integrity of prisms and ladders in [2] (Theorem 2.17 and Theorem 2.18 in [1]). We also compute the integrity of several special graphs.

We apply a lattice point counting method due to Blass and Sagan [2] to compute the characteristic polynomials for the subspace arrangements interpolated between the Coxeter hyperplane arrangements. Our proofs provide combinatorial interpretations for the characteristic polynomials of such subspace arrangements. In the process of doing this, we explore some interesting properties of these polynomials.

A graph of a puzzle is obtained by associating each possible position with a vertex and by inserting edges between vertices if and only if the corresponding positions can be obtained from each other in one move. Computational methods for finding the vertices at maximum distance \(\delta\) from a vertex associated with a goal position are presented. Solutions are given for small sliding block puzzles, and methods for obtaining upper and lower bounds on \(\delta\) for large puzzles are considered. Old results are surveyed, and a new upper bound for the 24-puzzle is obtained: \(\delta \leq 210\).

The total domination number \(\gamma_t(G)\) of graph \(G = (V, E)\) is the cardinality of a smallest subset \(S\) of \(V\) such that every vertex of \(V\) has a neighbor in \(S\). It is known that, if \(G\) is a connected graph with \(n\) vertices, \(\gamma_t(G) \leq \left\lfloor{2n}/{3}\right\rfloor\). Graphs achieving this bound are characterized.