If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+(x)\) and \(d^-(x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+(x),d^-(x)\} – \min\{d^+(y),d^-(y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)). If \(i_g(D) = 0\), then \(D\) is regular and if \(i_g(D) \leq 1\), then \(D\) is almost regular.

A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. It is easy to see that there exist regular \(c\)-partite tournaments with arbitrarily large \(c\) which contain arcs that do not belong to a directed cycle of length \(3\). In this paper we show, however, that every arc of an almost regular \(c\)-partite tournament is contained in a directed cycle of length four, when \(c \geq 8\). Examples show that the condition \(c \geq 8\) is best possible.

We address the following problem: What minimum degree forces a graph on \(n\) vertices to have a cycle with at least \(c\) chords? We prove that any graph with minimum degree \(\delta\) has a cycle with at least \(\frac{(\delta+1)(\delta-2)}{2}\) chords. We investigate asymptotic behaviour for large \(n\) and \(c\) and we consider the special case where \(n = c\).

We prove that a finite set \(A\) of points in the \(n\)-dimensional Euclidean space \(\mathcal{R}^n\) is uniquely determined up to translation by three of its subsets of cardinality \(|A|-1\) given up to translation, i.e. the Reconstruction Number of such objects is three. This result is best-possible.

We solve the problem of existence of minimal enclosings for triple systems with \(1 \leq \lambda \leq 6\) and any \(v\), i.e., an inclusion of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for minimal positive \(m\). A new necessary general condition is derived and some general results are obtained for larger \(\lambda\) values.

Colour the edges of a \(K_{24n+1}\) by \(12\) colours so that every vertex in every colour has degree \(2n\). Is there a totally multicoloured \(C_4\) (i.e. every edge gets a different colour)? Here we answer in the affirmative to this question. In [1] P. Erdős stated the same problem for \(K_{12n+1}\) and \(6\) colours, it was settled in [2].

In this paper we follow the terminology and symbols of [3]. We assume the complete graph \(K_{24n+1}\) to have the vertex-set \(V=V(K_{24n+1}) = \{1, 2, \ldots, 24n+1\}\).

Let \(P(G)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G) = P(H)\). A graph \(G\) is chromatically unique if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In “Chromatic Equivalence Classes of Certain Generalized Polygon Trees”, Discrete Mathematics Vol. \(172, 108–114 (1997)\), Peng \(et\; al\). studied the chromaticity of certain generalized polygon trees. In this paper, we present a chromaticity characterization of another big family of such graphs.

The step domination number of all graphs of diameter two is determined.

We use generator matrices \(G\) satisfying \(GG^T = aI + bJ\) over \(\mathbb{Z}_k\) to obtain linear self-orthogonal and self-dual codes. We give a new family of linear self-orthogonal codes over \(\text{GF}(3)\) and \(\mathbb{Z}_4\) and a new family of linear self-dual codes over \(\text{GF}(3)\).