We consider families of linear self-orthogonal and self-dual codes over the ring \({Z}_4\), which are generated by weighing matrices \(W(n, k)\) with \(k \equiv 0 \pmod{4}\), whose entries are interpreted as elements of the ring \({Z}_4\). We obtain binary formally self-dual codes of minimal Hamming distance 4 by applying the Gray map to the quaternary codes generated by \(W(n, 4)\).

Let \(G = (V, E)\) be a simple, undirected graph. A set of vertices \(D\) is called an odd dominating set if for every vertex \(v \in V(G)\), \(|N[v] \cap D| \equiv 1 \pmod{2}\). The minimum cardinality of an odd dominating set is called the odd domination number of \(G\). It is well known that every graph contains an odd dominating set, but this parameter has been studied very little. Our aim in this paper is to explore some basic features of the odd domination number and to compare it with the domination number of the graph, denoted by \(\gamma(G)\). In addition, extremal values of \(\gamma_{odd}(G)\) are calculated for several classes of graphs and a Nordhaus-Gaddum type inequality \(\gamma_{odd}(G) + \gamma_{odd}(\overline{G})\) is considered.

In this paper, it will be shown that a Skolem sequence of order \(n \equiv 0,1 \pmod{4}\) implies the existence of a graceful tree on \(2n\) vertices which exhibits a perfect matching or a matching on \(2n-2\) vertices. It will also be shown that a Hooked-Skolem sequence of order \(n \equiv 2,3 \pmod{4}\) implies the existence of a graceful tree on \(2n+1\) vertices which exhibits a matching on either \(2n\) or \(2n-2\) vertices. These results will be established using an algorithmic approach.

For \(k \geq 1\) an integer, a set \(D\) of vertices of a graph \(G = (V, E)\) is a \(k\)-dominating set of \(G\) if every vertex in \(V – D\) is within distance \(k\) from some vertex of \(D\). The \(k\)-domination number \(\gamma_k(G)\) of \(G\) is the minimum cardinality among all \(k\)-dominating sets of \(G\). For \(\ell \geq 2\) an integer, the graph \(G\) is \((\gamma_k, \ell)\)-critical if \(\gamma_k(G) = \ell\) and \(\gamma_k(G – v) = \ell – 1\) for all vertices \(v\) of \(G\). If \(G\) is \((\gamma_k, \ell)\)-critical for some \(\ell\), then \(G\) is also called a \(\gamma_k\)-critical graph. For a vertex \(v\) of \(G\), let \(N_k(v) = \{u \in V – \{v\} | d(u,v) \leq k\}\) and let \(\delta_k(G) = \min\{|N_k(v)|: v \in V\}\) and let \(\Delta_k(G) = \max\{|N_k(v)|: v \in V\}\). It is shown that if \(G\) is a nontrivial connected \(\gamma_k\)-critical graph, then \(\delta_k(G) \geq 2k\). Further, it is established that the number of vertices in a \(\gamma-k\)-critical graph \(G\) is bounded above by \((\Delta_k(G)+1)(\gamma_k(G)-1)+1\) and that \(G\) is a \((\gamma_k, \ell)\)-critical graph if and only if the \(k\)th power of \(G\) is a \((\gamma, \ell)\)-critical graph. It is shown that \((k, \ell)\)-critical graphs of arbitrarily large connectivity exist. Moreover, a graph without isolated vertices is shown to be \(\gamma_k\)-critical if and only if each of its blocks is \(\gamma_k\)-critical. Finally it is established that for an integer \(\ell \geq 2\), every graph is an induced subgraph of some \((\gamma_k, \ell)\)-critical graph. This paper concludes with some partially answered questions and some open problems.

We provide complete lists of starters and Skolem sequences which generate perfect one-factorizations of complete graphs up to order \(32\) for starters and \(36\) for Skolem sequences. The resulting perfect one-factorizations are grouped into isomorphism classes, and further analysis of the results is performed.

We find new full orthogonal designs in order 72 and show that of 2700 possible \(OD(72; s_1, s_2, s_3, 72 – s_1 – s_2 – s_3)\), 335 are known, of 432 possible \(OD(72; s_1, s_2, 72 – s_1 – s_2)\), 308 are known. All possible \(OD(72; s_1, 72 – s_1)\) are known.

Classical bin packing has been studied extensively in the literature. Open-ends bin packing is a variant of the classical bin packing. Open-ends bin packing allows pieces to be partially beyond a bin, while the classical bin packing requires all pieces to be completely inside a bin. We investigate the open-ends bin packing problem for both the off-line and on-line versions and give algorithms to solve the problem for parametric cases.

The queen’s graph \(Q_n\) has the squares of the \(n \times n\) chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let \(\gamma(Q_n)\) be the minimum size of a dominating set of \(Q_n\). Spencer proved that \(\gamma(Q_n) \geq {(n-1)}/{2}\) for all \(n\), and the author showed \(\gamma(Q_n) = {(n-1)}/{2}\) implies \(n \equiv 3 \pmod{4}\) and any minimum dominating set of \(Q_n\) is independent.

Define a sequence by \(n_1 = 3\), \(n_2 = 11\), and for \(i > 2\), \(n_i = 4n_{i-1} – n_{i-2} – 2\). We show that if \(\gamma(Q_n) = {(n-1)}/{2}\) then \(n\) is a member of the sequence other than \(n_3 = 39\), and (counting from the center) the rows and columns occupied by any minimum dominating set of \(Q_n\) are exactly the even-numbered ones. This improvement in the lower bound enables us to find the exact value of \(\gamma(Q_n)\) for several \(n\); \(\gamma(Q_n) = {(n+1)}/{2}\) is shown here for \(n = 23, 39\), and elsewhere for \(n = 27, 71, 91, 115, 131\).

A characterization of symmetric bent functions has been presented in [3]. Here, we provide a simple proof of the same result.

We prove that the total domination number of an \(n\)-vertex claw-free cubic graph is at most \({n}/{2}\).