Let \(G\) and \(H\) be connected graphs and let \(G \square H\) be the Cartesian product of \(G\) by \(H\). A lower and an upper bound for the independence number of the Cartesian product of graphs is proved for the case, where one of the factors is bipartite. Cartesian products with one factor being an odd path or an odd cycle are considered as well.

It is proved in particular that if \(S_1 + S_2\) is a largest 2-independent set of a graph \(G\), such that \(|S_2|\) is as small as possible and if \(|S_2| \leq n+2\), then \(\alpha(G \square P_{2n+1}) = (n+1)|S_1| + n|S_2|\). A similar result is shown for the Cartesian product with an odd cycle. It is finally proved that \(\alpha(C_{2k+1} \square C_{2n+1}) = k(2n+1)\), extending a result of Jha and Slutzki.

Parallel processing has been a valuable tool for improving the performance of many algorithms. Solving intractable problems is an attractive application of parallel processing. Traditionally, exhaustive search techniques have been used to find solutions to \(NP\)-complete problems. However, the performance benefit of parallelization of exhaustive search algorithms can only provide linear speedup, which is typically of little use as problem complexity increases exponentially with problem size. Genetic algorithms can be useful tools to provide satisfactory results to such problems. This paper presents a genetic algorithm that uses parallel processing in a cooperative fashion to determine mappings for the rectilinear crossing problem. Results from this genetic algorithm are presented which contradict a conjecture that has been open for over 20 years regarding the minimal crossing number for rectilinear graphs.

A balanced tournament design, \(\mathrm{BTD}(n)\), defined on a \(2n\)-set \(V\), is an arrangement of the \(\binom{2n}{2}\) distinct unordered pairs of the elements of \(V\) into an \(n \times 2n-1\) array such that:
(1) every element of \(V\) is contained in precisely one cell of each column, and
(2) every element of \(V\) is contained in at most two cells of each row.

If we can partition the columns of a \(\mathrm{BTD}(n)\) defined on \(V\) into three sets \(C_1, C_2, C_3\) of sizes \(1, n-1, n-1\) respectively such that the columns in \(C_1 \cup C_2\) form a Howell design of side \(m\) and order \(2n\), an \(\mathrm{H}(n,2n)\), and the columns in \(C_1 \cup C_3\) form an \(\mathrm{H}(n,2n)\), then the \(\mathrm{BTD}(n)\) is called partitionable. We denote a partitioned balanced tournament design of side \(n\) by \(\mathrm{PBTD}(n)\). The existence of these designs has been determined except for seven possible exceptions. In this note, we describe constructions for four of these designs. This completes the spectrum of \(\mathrm{PBTD}(n)\) for \(n\) even.

In this note we complete the table of Ramsey numbers for \(K_s\) versus the family of all \((p,q)\)-graphs for \(p \leq 8\).
Moreover, some results are obtained for the general case.

Let \(G\) be a \(2\)-connected graph of order \(n\) with connectivity \(\kappa\) and independence number \(\alpha\). In this paper, we show that if for each independent set \(S\) with \(|S| = k+1\), there are \(u,v \in S\) such that satisfying one of the following conditions:

1. \(d(u) + d(v) \geq n\); or \(|N(u) \cap N(v)| \geq \alpha\); or \(|N(u) \cup N(v)| \geq n-k\);
2. for any \(x \in \{u,v\}\), \(y \in V(G)\) and \(d(x,y) = 2\), it implies that \(\max\{d(x), d(y)\} \geq n/2\),

then \(G\) is hamiltonian. This result reveals the internal relations among several well-known sufficient conditions: \((1)\) it shows that it does not need to consider all pairs of nonadjacent or distance two vertices in \(G\); \((2)\) it makes known that for different pairs of vertices in \(G\), it permits to satisfy different conditions.

Let \(G\) be a graph of order \(p\) such that both \(G\) and \(\overline{G}\) have no isolated vertices. Let \(\Upsilon_t\) and \(\overline{\Upsilon}_t\) denote respectively the total domination number of \(G\) and \(\overline{G}\). In this paper we obtain a characterization of all graphs \(G\) for which \\(i) \(\Upsilon_t +\overline{\Upsilon}_t= p+1\) and (ii) \(\Upsilon_t + \overline{\Upsilon}_t = p\).

The bondage number \(b(G)\) of a nonempty graph \(G\) was first introduced by Fink, Jacobson, Kinch, and Roberts [3]. In their paper they conjectured that \(b(G) \leq \Delta(G)+1\) for a nonempty graph \(G\). A counterexample for this conjecture was shown in [5]. Beyond it, we show now that there doesn’t exist an upper bound of the following form: \(b(G) \leq \Delta(G)+c\) for any \(c\in\mathbb{N}\).

It is shown that if \(t > 1\) and \(u \geq 5\), then the known necessary condition for the existence of a skew Room frame of type \(t^u\), is also sufficient with the possible exception of \((u, f)\) where \(u = 5\) and \(t \in \{11, 13, 17, 19, 23, 29, 31, 41, 43\}\).

The class of \(t-sc\) graphs constitutes a new generalization of self-complementary graphs. Many \(t-sc\) graphs exhibit a stable complementing permutation. In this paper, we prove a sufficient condition for the existence of a stable complementing permutation in a \(t-sc\) graph. We also construct several infinite classes of \(t-sc\) graphs to show the stringency of our sufficient condition.

A polyhex graph is either a hexagonal system or a coronoid system. A polyhex graph \(G\) is said to be \(k\)-coverable if for any \(k\) mutually disjoint hexagons the subgraph obtained from \(G\) by deleting all these \(k\) hexagons together with their incident edges has at least one perfect matching. In this paper, a constructive criterion is given to determine whether or not a given polyhex graph is \(k\)-coverable. Furthermore, a simple method is developed which allows us to determine whether or not there exists a \(k\)-coverable polyhex graph with exactly \(h\) hexagons.