Let \(v, k,\lambda\) and \(n\) be positive integers. \((x_1, x_2, \ldots, x_k)\) is defined to be \(\{(x_i, x_j) : i \neq j, i,j =1,2,\ldots,k\},\) in which the ordered pair \((x_i, x_j)\) is called \((j-i)\)-apart for \(i > j\) and \((k+j-i)\)-apart for \(i > j\), and is called a cyclically ordered \(k\)-subset of \(\{x_1, x_2, \ldots, x_k\}\).

A perfect Mendelsohn design, denoted by \((v, k, \lambda)\)-PMD, is a pair \((X, B)\), where \(X\) is a \(v\)-set (of points), and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks), such that every ordered pair of points of \(X\) appears \(t\)-apart in exactly \(\lambda\) blocks of \(B\) for any \(t\), where \(1 \leq t \leq k-1\).

If the blocks of a \((v, k, \lambda)\)-PMD for which \(v \equiv 0 \pmod{k}\) can be partitioned into \(\lambda(v-1)\) sets each containing \(v/k\) blocks which are pairwise disjoint, the \((v, k, \lambda)\)-PMD is called resolvable, denoted by \((v, k, \lambda)\)-RPMD.

In the paper [14], we have showed that a \((v, 4, 1)\)-RPMD exists for all \(v \equiv 0 \pmod{4}\) except for \(4, 8\) and with at most \(49\) possible exceptions of which the largest is \(336\).

In this article, we shall show that a \((v, 4, 1)\)-RPMD for all \(v \equiv 0 \pmod{4}\) except for \(4, 8, 12\) and with at most \(27\) possible exceptions of which the largest is \(188\).

The independence number of Cartesian product graphs is considered. An upper bound is presented that covers all previously known upper bounds. A construction is described that produces a maximal independent set of a Cartesian product graph and turns out to be a reasonably good lower bound for the independence number. The construction defines an invariant of Cartesian product graphs that is compared with its independence number. Several exact independence numbers of products of bipartite graphs are also obtained.

Multi-loop digraphs are widely studied mainly because of their symmetric properties and their applications to loop networks. A multi-loop digraph, \(G = G(N; s_1, \ldots, s_\Delta)\) with \(1 \leq s_1 < \cdots < s_\Delta \leq N-1\) and \(\gcd(N, s_1, \ldots, s_\Delta) = 1\), has set of vertices \(V ={Z}_N\) and adjacencies given by \(v \mapsto v + s_i \mod N, i = 1, \ldots, \Delta\). For every fixed \(N\), an usual extremal problem is to find the minimum value \[D_\Delta(N)=\min\limits_{s_1,\ldots,s_\Delta \in Z_N}(N; s_1, \ldots, s_\Delta)\] where \(D(N; s_1, \ldots, s_\Delta)\) is the diameter of \(G\). A closely related problem is to find the maximum number of vertices for a fixed value of the diameter. For \(\Delta = 2\), all optimal families have been found by using a geometrical approach. For \(\Delta = 3\), only some dense families are known. In this work, a new dense family is given for \(\Delta = 3\) using a geometrical approach. This technique was already adopted in several papers for \(\Delta = 2\) (see for instance [5, 7]). This family improves the dense families recently found by several authors.

Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1 (1999), 387-400) considered a variation of the classical Turén-type extremal problems as follows: for a given graph \(H\), determine the smallest even integer \(\sigma (H,n)\) such that every \(n\)-term positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(H,n)\) has a realization \(G\) containing \(H\) as a subgraph. In particular, they pointed out that \(3n – 2 \leq \sigma(K_{4} – e, n) \leq 4n – 4\), where \(K_{r+1} – e\) denotes the graph obtained by removing one edge from the complete graph \(K_{r+1}\) on \(r+1\) vertices. Recently, Lai determined the values of \(\sigma(K_4 – e, n)\) for \(n \geq 4\). In this paper, we determine the values of \(\sigma(K_{r+1} – e, n)\) for \(r \geq 3\) and \(r+1 \leq n \leq 2r\), and give a lower bound of \(\sigma(K_{r+1} – e, n)\). In addition, we prove that \(\sigma(K_5 – e, n) = 5n – 6\) for even \(n\) and \(n \geq 10\) and \(\sigma(K_5 – e, n) = 5n – 7\) for odd \(n\) and \(n \geq 9\).

We show that if \(G\) is a \(3\)-connected graph of order at least \(5\), then there exists a longest cycle \(C\) of \(G\) such that the number of contractible edges of \(G\) which are on \(C\) is greater than or equal to \(\frac{|V(C)| + 9}{8}.\)

We obtain lower bounds for the number of elements dominated by a subgroup in a Cayley graph. Let \(G\) be a finite group and let \(U\) be a generating set for \(G\) such that \(U = U^{-1}\) and \(1 \in U\). Let \(A\) be an independent subgroup of \(G\). Let \(r\) be a positive integer, and suppose that, in the Cayley graph \((G,U)\), any two non-adjacent vertices have at most \(r\) common neighbours. Let \(N[H]\) denote the set of elements of \(G\) which are dominated by the elements of \(H\). We prove that

1. \(|N[A]| \geq \lceil\frac{|U|^2}{r(|H \cup U^2|-1)+|U|}\rceil.\)
2. \(|N[A]| \geq |U||H| -\frac{|H|r}{2}(|H \cup U^2|-1).\)

An interesting example illustrating these results is the graph on the symmetric group \(S_n\), in which two permutations are adjacent if one can be obtained from the other by moving one element. For this graph we show that \(r = 4\) and illustrate the inequalities.

Shapiro [8] asked what simple family of circuits will have resistances \(C_{2n}/{C_{2n}-1}\) (or something similar) where \(C_m=\frac{1}{m+1}\binom{2m}{m}\) is the \(m\)th Catalan number. In this paper, we give a construction of such circuits; we also discuss some related problems.

The two-dimensional bandwidth problem is to determine an embedding of graph \(G\) in a grid graph in the plane such that the longest edges are as short as possible. In this paper, we study the problem under the distance of \(L_\infty\)-norm.

Domination graphs of directed graphs have been defined and studied in a series of papers by Fisher, Lundgren, Guichard, Merz, and Reid. A tie in a tournament may be represented as a double arc in the tournament. In this paper, we examine domination graphs of tournaments, tournaments with double arcs, and more general digraphs.