The concept of a strong \(a\)-valuation was introduced by Maheo, who showed that if a graph \(G\) has a strong \(a\)-valuation, then so does \(G \times K_2\). We show that for various graphs \(G\), \(G \times Q_n\) has a strong \(a\)-valuation and \(G \times P_n\) has an \(a\)-valuation, where \(Q_n\) is the \(n\)-cube and \(P_n\) the path with \(n\) edges, including \(G = K_{m,2}\) for any \(m\). Yet we show that \(K_{m,n} \times K_2\) does not have a strong \(a\)-valuation if \(m\) and \(n\) are distinct odd integers.

Let \(p\) be an odd prime number. We introduce a simple and useful decoding algorithm for orthogonal Latin square codes of order \(p\). Let \({H}\) be the parity check matrix of orthogonal Latin square code. For any \({x} \in {GF}(p)^n\), we call \(2 {H}^t\) the syndrome of \({x}\). This method is based on the syndrome-distribution decoding for linear codes. In \(\mathcal {L}_p\), we need to find the first and the second coordinates of codeword in order to correct the errored received vector.

The maximum cardinality of a partition of the vertex set of a graph \(G\) into dominating sets is the domatic number of \(G\), denoted \(d(G)\). We consider Nordhaus-Gaddum type results involving the domatic number of a graph, where a Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. Thereafter we investigate the upper bounds on the sum and product of the domatic numbers \(d(G_1), d(G_2)\) and \(d(G_3)\) where \(G_1 \oplus G_2 \oplus G_3 = K_n\). We show that the upper bound on the sum is \(n+2\), while the maximum value of the product is \(\lceil \frac{n}{3} \rceil ^3\) for \(n > 57\).

Place a checker in some square of an \(n \times n\) checkerboard. The checker is allowed to step either to the east or to the north, and is allowed to step off the edge of the board in a manner suggested by the usual identification of the edges of the square to form a projective plane. We give an explicit description of all the routes that can be taken by the checker to visit each square exactly once.

Bailey (1989) defined a \(k \times v\) double Youden rectangle (DYR), with \(k 3\) is a prime power with \(k \equiv 3 \pmod{4}\). We now provide a general construction for DYRs of sizes \(k \times (2k+1)\) where \(k > 5\) is a prime power with \(k \equiv 1 \pmod{4}\). We present DYRs of sizes \(9 \times 19\) and \(13 \times 27\).

We show by an elementary argument that, given any greedy clique decomposition of a graph \(G\) with \(n\) vertices, the sum of the orders of the cliques is less than \(\frac{5}{8}n^2\). This gives support to a conjecture of Peter Winkler.

We study the signed domination number \(\gamma_s\), the minus domination number \(\gamma^-\) and the majority domination number \(\gamma_{\mathrm{maj}}\). In this paper, we establish good lower bounds for \(\gamma_s\), \(\gamma^-\) and \(\gamma_{\mathrm{maj}}\), and give sharp lower bounds for \(\gamma_s\), \(\gamma^-\) for trees.

In this paper, nineteen new binary linear codes are presented which improve the bounds on the maximum possible minimum distance. These codes belong to the class of quasi-cyclic (QC) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Six of the new codes meet the upper bound on minimum distance and so are optimal.

This game is a mixture of Searching and Cops and Robber. The Cops have partial information provided by sensing devices called photo radar. The Robber has perfect information. We give bounds on the number of photo radar units required by one Cop to capture a Robber on a tree and, with less tight bounds, on a copwin graph.

Cographs—complement-reducible graphs—can be viewed as intersection graphs (of \(k\)-dimensional boxes), as intersections of graphs (of \(P_4 ,C_4\)-free graphs), and as common tieset graphs of two-terminal graphs. This approach connects cographs with other topics such as chordal, interval, and series-parallel graphs, and it provides a natural dimension for cographs.