We prove: A connected magic graph with \(n\) vertices and \(q\) edges exists if and only if \(n = 2\) and \(q = 1\) or \(n \geq 5\) and \(\frac{5n}{4} < q < \frac{n(n-1)}{2} \).

Sharp bounds are presented for the \(\lambda\)-number of the Cartesian product of a cycle and a path, and of the Cartesian product of two cycles.

A set \(S = \{v_1, v_2, \ldots, v_n\}\) of vertices in a graph \(G\) with associated sequence \(k_1, k_2, \ldots, k_n\) of nonnegative integers is called a step domination set if every vertex of \(G\) is at distance \(k_i\) from \(v_i\) for exactly one \(i\) (\(1 \leq i \leq n\)). The minimum cardinality of a step domination set is called the step domination number of \(G\). This parameter is determined for several classes of graphs and is investigated for trees.

We completely determine the spectrum (i.e. set of orders) of complete \(4\)-partite graphs with at most one odd part which are decomposable into two isomorphic factors with a finite diameter. For complete \(4\)-partite graphs with all parts odd we solve the spectrum problem completely for factors with diameter \(5\). As regards the remaining possible finite diameters, \(2, 3, 4\), we present partial results, focusing on decompositions of \(K_{n,n,n,m}\) and \(K_{n,n,m,m}\) for odd \(m\) and \(n\).

In this paper we determine the \(k\)-domination numbers of the cardinal products \(P_2 \times P_n, \ldots, P_{2k+1} \times P_n\) for all integers \(k \geq 2, n \geq 3\).

In this paper we investigate the nature of both the \(2\)-packing number and the minimum domination number of the cartesian product of graphs where at least one of them has the property that every vertex is either a leaf or has at least one leaf as a neighbour.

Let \(H\) be a graph, and let \(k\) be a positive integer. A graph \(G\) is \(H\)-coverable with overlap \(k\) if there is a covering of all the edges of \(G\) by copies of \(H\) such that no edge of \(G\) is covered more than \(k\) times. The number \(ol(H, G)\) is the minimum \(k\) for which \(G\) is \(H\)-coverable with overlap \(k\).

It is established (Theorem 2.1) that if \(n\) is sufficiently large then
\[ol(H, K_n) \leq 2.\]

For \(H\) being a path, a matching or a star it is enough to assume \(|H| \leq n\) (Theorem 3.1).

The same result is obtained (Main Theorem) for any graph \(H\) having at most four vertices, or else at most four edges with a single exception \(ol(K_4, K_5) = 3\).

Given a digraph (an undirected graph, resp.) \(D\) and two positive integers \(f(x), g(x)\) for every \(x \in V(D)\), a subgraph \(H\) of \(D\) is called a \((g, f)\)-factor if \(g(x) \leq d^+_H(x) = d^-_H(x) \leq f(x)\) (\(g(x) \leq d_H(x) \leq f(x)\), resp.) for every \(x \in V(D)\). If \(f(x) = g(x) = 1\) for every \(x\), then a connected \((g, f)\)-factor is a hamiltonian cycle. The previous research related to the topic has been carried out either for \((g, f)\)-factors (in general, disconnected) or for hamiltonian cycles separately, even though numerous similarities between them have been recently detected. Here we consider connected \((g, f)\)-factors in digraphs and show that several results on hamiltonian digraphs, which are generalizations of tournaments, can be extended to connected \((g, f)\)-factors. Applications of these results to supereulerian digraphs are also obtained.