We derive a formula for the expected value \(\mu(2n+1)\) of the independent domination number of a random binary tree with \(2n+1\) vertices and determine the asymptotic behavior of \(\mu(2n+1)\) as \(n\) goes to infinity.

In [5], Gueizow gave an example of semiboolean SQS-skeins of nilpotent class \(2\), all its derived sloops are Boolean “or” of nilpotence class \(1\). In this paper, we give an example of nilpotent SQS-skein of class \(2\) whose derived sloops are all of nilpotence class \(2\). Guelzow [6] has also given a construction of semiboolean SQS-skeins of nilpotence class \(n\) whose derived sloops are all of class \(1\). As an extension result, we prove in the present paper the existence of nilpotent SQS-skeins of class \(n\) all of whose derived sloops are nilpotent of the same class \(n\); for any positive integer \(n\).

In this note we solve almost completely a problem raised by Topp and Volkmann [7] concerning the product of the domination and the chromatic numbers of a graph.

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.