Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A vertex labelling \( f: V \rightarrow \{0, 1, 2\} \) induces an edge labelling \( \overline{f}: E \rightarrow \{0, 1, 2\} \) defined by \( \overline{f}(uv) = |f(u) – f(v)| \). Let \( v_f(0), v_f(1), v_f(2) \) denote the number of vertices \( v \) with \( f(v) = 0, f(v) = 1 \) and \( f(v) = 2 \) respectively. Let \( e_f(0), e_f(1), e_f(2) \) be similarly defined. A graph is said to be 3-equitable if there exists a vertex labelling \( f \) such that \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for \( 0 \leq i, j \leq 2 \). In this paper, we show that every multiple shell \( MS\{n_1^{t_1}, \ldots, n_r^{t_r}\} \) is 3-equitable for all positive integers \( n_1, \ldots, n_r, t_1, \ldots, t_r \).

The rainbow Ramsey number \( RR(G_1, G_2) \) or constrained Ramsey number \( f(G_1,G_2) \) of two graphs \( G_1 \) and \( G_2 \) is defined to be the minimum integer \( N \) such that any edge-coloring of the complete graph \( K_N \) with any number of colors must contain either a subgraph isomorphic to \( G_1 \) with every edge the same color or a subgraph isomorphic to \( G_2 \) with every edge a different color. This number exists if and only if \( G_1 \) is a star or \( G_2 \) is acyclic. In this paper, we present the conjecture that the constrained Ramsey number of \( nK_2 \) and \( mK_2 \) is \( m(n-1)+2 \), along with a proof in the case \( m \leq \frac{3}{2}(n-1) \).

A set \( C \subseteq \mathbb{F}_2^n \) is said to be an asymmetric covering code with radius \( R \) if every word \( x \in \mathbb{F}_2^n \) can be obtained by replacing \( 1 \) by \( 0 \) in at most \( R \) coordinates of a word in \( C \). In this paper, tabu search is employed in the search for good asymmetric covering codes of small length. Fifteen new upper bounds on the minimum size of such codes are obtained in the range \( n \leq 13 \).

We use exhaustive computer searches to show that there are exactly \( 36 \) codewords in an optimal ternary \( (11,7) \) code and exactly \( 13 \) codewords in an optimal ternary \( (14,10) \) code. We also enumerate inequivalent optimal ternary \( (14,10) \) codes and show that there are exactly \( 6151 \) such codes.

The minimum number of blocks having maximum size precisely four that are required to cover, exactly \( \lambda \) times, all pairs of elements from a set of cardinality \( v \) is denoted by \( g_\lambda^{(4)}(v) \). We present a complete solution to this problem for \( v = 3, 4, \) and \( 5 \).

Among the well-studied maximal planar graphs, those having the maximum possible number of 3-cycles are precisely the planar chordal graphs (meaning no induced cycles of lengths greater than three). This motivates a somewhat similar result connecting maximal planar bipartite graphs, 4-cycles, and planar chordal bipartite graphs (meaning bipartite with no induced cycles of lengths greater than four), together with characterizations of planar chordal bipartite graphs as radial graphs of outerplanar multigraphs.

Combinatorial designs are a powerful tool because of their beautiful combinatorial structure that can help in many applications, such as coding theory or cryptography. A conference key distribution system is a scheme to design a conference key, and then to distribute this key to only participants attending the conference in order to communicate with each other securely. In this paper, we present an efficient conference key distribution system using difference families. Using techniques for creating the conference key and for performing authentication based on identification information, the communication protocol is designed. Applying the known results on difference families, we obtain many new infinite classes of conference key distribution systems. In special classes of difference families, the message overhead is \( O(v\sqrt{tv}) \), where \( v \) is the number of participants and \( t \) is the number of the \( k \)-elements subsets that consist of the difference family. The security of the presented protocol, which is an important problem in the construction of a secure system, is proved to be as computationally difficult to calculate as factoring and discrete logarithms.

In this paper, finite \( \{2,t\} \)-semiaffine linear spaces are investigated. When \( t = 5 \), their parameters are determined, and it is also proved that there is a single finite \( \{2, 5\} \)-semiaffine linear space on \( v = 20 \) points and with constant point degree \( 7 \).

We establish that for each of the 5005 possible types of 2-factorizations of the complete graph \( K_{13} \), there exists at least one solution. We also enumerate all nonisomorphic solutions to the Oberwolfach problem \( \text{OP}(13;3,3,3,4) \).

Scheduling static tasks on parallel architectures is a basic problem arising in the design of parallel algorithms. This NP-complete problem has been widely investigated in the literature and remains one of the most challenging questions in the field. Among the resolution methods for this type of problems, the taboo search technique is of particular interest. Based on this technique, two algorithms are proposed and tested on a sample of instances in order to be compared experimentally with other well-known algorithms. The results clearly indicate good overall performances of our algorithms. Next, some NP-completeness results are established showing that this problem is intractable for approximation, even for some restricted cases bearing a clear relation to the instances treated experimentally in this work.