Using path counting arguments, we prove
\(min\{\binom{x_1+x_2+y_1+y_2}{x_1,x_2,(y_1+y_2)},\binom{(x_1+x_2+y_1+y_2)}{(x_1+x_2),y_1,y_2}\}\leq\binom{x_1+y_1}{x_1}\binom{x_1+y_2}{x_1}\binom{x_2+y_1}{x_2}\binom{x_2+y_2}{x_2}\)

This inequality, motivated by graph coloring considerations, has an interesting geometric interpretation.

The existence of holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOMs) of types \(h^n\) and \(1^{n}u^1\) is investigated. For type \(h^n\), new pairs of \((h, n)\) are constructed so that the possible exceptions of \((h, n)\) for the existence of such HSOLSSOMs are reduced to \(11\) in number. Two necessary conditions for the existence of HSOLSSOMs of type \(1^{n}u^1\) are (1) \(n \geq 3u + 1\) and (2) \(n\) must be even and \(u\) odd. Such an HSOLSSOM gives rise to an incomplete SOLSSOM. For \(3 \leq u \leq 15\), the necessary conditions are shown to be sufficient with seven possible exceptions. It is also proved that such an HSOLSSOM exists whenever even \(n \geq 5u + 9\) and odd \(u \leq 9\).

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 \(ol(K_4, K_5) = 3\).