A transitive triple, \((a, b, c)\), is defined to be the set \(\{(a, b), (b, c), (a, c)\}\) of ordered pairs. A directed triple system of order \(v\), \(DTS(v)\), is a pair \((D, \beta)\), where \(D\) is a set of \(v\) points and \(\beta\) is a collection of transitive triples of pairwise distinct points of \(D\) such that any ordered pair of distinct points of \(D\) is contained in precisely one transitive triple of \(\beta\). An antiautomorphism of a directed triple system, \((D, \beta)\), is a permutation of \(D\) which maps \(\beta\) to \(\beta^{-1}\), where \(\beta^{-1} = \{(c, b, a) | (a, b, c) \in \beta\}\). In this paper, we give necessary and sufficient conditions for the existence of a directed triple system of order \(v\) admitting an antiautomorphism consisting of two cycles of lengths \(M\) and \(2M\), and one fixed point.

A \(k\)-circuit is a directed cycle of length \(k\). In this paper, we completely solve the problem of finding maximum packings and minimum coverings of \(\lambda\)-fold complete bipartite symmetric digraphs with \(6\)-circuits.

Until now, all known simple \(t-(v, k, \lambda)\) designs with \(t \geq 6\) have \(\lambda \geq 4\). On the other hand, P. J. Cameron and C. E. Praeger showed that there are no flag-transitive simple \(7-(v, k, \lambda)\) designs. In the present paper, we considered the flag-transitive simple \(6-(v, k, \lambda)\) designs and proved that there are no non-trivial flag-transitive simple \(6-(v, k, \lambda)\) designs with \(\lambda \leq 5\).

In \(1972\), Erdős, Faber, and Lovász made the now famous conjecture: If a graph \(G\) consists of \(n\) copies of the complete graph \(K_n\), such that any two copies have at most one common vertex (such graphs are called EFL graphs), then \(G\) is \(n\)-colorable. In this paper, we show that the conjecture is true for two different classes of EFL graphs. Furthermore, a new shorter proof of the conjecture is given for a third class of EFL graphs.

The edge set of \(K_n\) cannot be decomposed into edge-disjoint octagons (or \(8\)-cycles) when \(n \not\equiv 1 \pmod{16}\). We consider the problem of removing edges from the edge set of \(K_n\) so that the remaining graph can be decomposed into edge-disjoint octagons. This paper gives the solution of finding maximum packings of complete graphs with edge-disjoint octagons and the minimum leaves are given.

In this paper, we introduced the notion of symmetric \(f\) bi-derivations on lattices and investigated some related properties. We characterized the distributive lattice by symmetric \(f\) bi-derivations.

In \(1990\), Anderson et al. \([1]\) generalized the competition graph of a digraph to the competition multigraph of a digraph and defined the multicompetition number of a multigraph. The competition multigraph \(CM(D)\) of a digraph \(D = (V, A)\) is the multigraph \(M = (V, E’)\) where two vertices of \(V\) are joined by \(k\) parallel edges if and only if they have exactly \(\ell\) common preys in \(D\). The multicompetition number \(k^*(M)\) of the multigraph \(M\) is the minimum number \(p\) such that \(M \cup I_p\) is the competition multigraph of an acyclic digraph, where \(I_k\) is a set of \(p\) isolated vertices. In this paper, we study the multicompetition numbers for some multigraphs and generalize some results provided by Kim and Roberts \([9]\), and by Zhao and He \([18]\) on general competition graphs, respectively.

Frank Harary contributed numerous questions to a variety of topics in graph theory. One of his favourite topics was the Reconstruction Problem, which, in its first issue in \(1977\), the Journal of Graph Theory described as the major unsolved problem in the field. Together with Plantholt, Frank Harary initiated the study of reconstruction numbers of graphs. We shall here present a survey of some of the work done on reconstruction numbers, focusing mainly on the questions which this work leaves open.