The cycle graph \(C(H)\) of a graph \(H\) is the edge intersection graph of all induced chordless cycles of \(H\). We investigate iterates of the mapping \(\overline{C}: G \rightarrow C(\overline{G})\) where \(C\) denotes the map that associates to a graph its cycle graph. We call a graph \(G\) vanishing under \(\overline{C}\) if \(\overline{C^n}(G) = 0\) for some \(n\), otherwise \(G\) is called \(\overline{C}\)-persistent. We call a graph \(G\) expanding under \(\overline{C}\) if \(|\overline{C^n}(G)| \to \infty\) as \(n \to \infty\). We show that the lowest order of a \(\overline{C}\)-expanding graph is \(6\) and determine the behaviour under \(\overline{C}\) of some special graphs, including trees, null graphs, cycles and complete bipartite graphs.

Nonbinary power residue codes are constructed using the relationship between these codes and quasi-cyclic codes. Eleven of these codes exceed the known lower bounds on the maximum possible minimum distance of a linear code.

In this paper the authors study one- and two-dimensional color switching problems by applying methods ranging from linear algebra to parity arguments, invariants, and generating functions. The variety of techniques offers different advantages for addressing the existence and uniqueness of minimal solutions, their characterizations, and lower bounds on their lengths. Useful examples for reducing problems to easier ones and for choosing tools based on simplicity or generality are presented. A novel application of generating functions provides a unifying treatment of all aspects of the problems considered.

Broadcasting refers to the process of information dissemination in a communication network whereby a message is to be sent from a single originator to all members of the network, subject to the restriction that a member may participate in only one message transfer during a given time unit. In this paper we present a family of broadcasting schemes over the odd graphs, \(O_{n+1}\). It is shown that the broadcast time of \(O_{n+1}\), \(b(O_{n+1})\), is bounded by \(2n\). Moreover, the conjecture that \(b(O_{n+1}) = 2n\) is put forward, and several facts supporting this conjecture are given.

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 \(n > 57\).