We present a permutation group whose orbits classify isomorphism of covering projections of the complete graph with \(4\) vertices. Two structure theorems concerning this group are proved.

Constructions have been completed which improve the lower bounds for \(R(4,6)\), \(R(5,6)\) and \(R(3,12)\).

Let \(G\) be a graph with minimum degree \(\delta\). For each \(i = 1, 2, \ldots, \delta \), let \(a_i(G)\) (resp. \(\alpha^*_i(G)\)) denote the smallest integer \(k\) such that \(G\) has an \([i, k]\)-factor (resp. a connected \([i, k]\)-factor). Denote by \(G_n\) a complete \(n\)-partite graph. In this paper, we determine the value of \(\alpha_t(G_n)\), and show that \(0 \leq \alpha^*_1(G_n) – \alpha(G_n) \leq 1\) and \(\alpha^*_i(G_n) = a_i(G_n)\) for each \(i = 2, 3, \ldots, \delta\).

An oriented (or ordered) triple means either a Mendelsohn or a transitive triple. An oriented (or ordered) triple system of order \(v\), briefly OTS(\(v\)), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of oriented triples of elements of \(V\), such that every ordered pair of distinct elements of \(V\) belongs to exactly one member of \(B\). It is known that an OTS(\(v\)) exists if and only if \(v \equiv 0, 1 \pmod{3}\). An OTS(\(v\)) is cyclic if it admits an automorphism consisting of a single cycle of length \(v\); an OTS(\(v\)) is rotational if it admits an automorphism consisting of a single fixed point and one cycle of length \(v-1\). In this note we give some constructions of OTS(\(v\))’s which allow to determine the spectrum of cyclic and of rotational OTS(\(v\))’s.

It is shown that the maximal number of pairwise edge disjoint trees of order seven in the complete graph \(K_n\), and the minimum number of trees of order seven, whose union is \(K_n\), are \(\left\lfloor\frac{n(n-1)}{12}\right\rfloor\) and \(\left\lceil\frac{n(n-1)}{12}\right\rceil,n\geq 11\), respectively. (\(\lfloor x\rfloor\) denotes the largest integer not exceeding \(x\) and \(\lceil x\rceil\) the least integer not less than \(x\)).

It is unknown whether or not there exists a \([51, 5, 33; 3]\)-code (meeting the Griesmer bound). The purpose of this paper is to show that there is no \([51, 5, 33; 3]\)-code.

The Hitting Set problem is investigated in relation to restrictions imposed on the cardinality of subsets and the frequency of element occurences in the subsets. It is shown that the Hitting Set subproblem where each subset has cardinality \(C\) for fixed \(C \geq 2\) and the frequency of each element is exactly \(f\) for fixed \(f \geq 3\) remains NP-complete, but the problem becomes polynomial when \(f \leq 2\). The restriction of the Vertex Cover problem to \(f\)-regular graphs for \(f \geq 3\) remains NP-complete.

Hill and Newton showed that there exists a \([20, 6, 12; 3]\)-code, and that the weight distribution of a \([20,5, 12; 3]\)-code is unique. However, it is unknown whether or not a code with these parameters is unique. Recently, Hamada and Helleseth showed that a \([19, 4, 12; 3]\)-code is unique up to equivalence, and characterized this code using a characterization of \(\{21, 6; 3, 3\}\)-minihypers. The purpose of this paper is to show, using the geometrical structure of the \([19, 4, 12; 3]\)-code, that exactly two non-isomorphic \([20, 5, 12; 3]\)-codes exist.

We obtain a new characterization, by a configuration theorem, of the Miquelian geometries among the finite inversive (= Möbius) planes of even order. The main tool used is a characterization due to J. Tits of elliptic ovoids in three-dimensional projective space,

Let \(E_n\) denote the minimum number of edges in a graph that contains every tree with \(n\) edges. This article provides two sets of data concerning \((n+1)\)-vertex graphs with \(E_n\) edges for each \(n \leq 11\): first, a minimum set of trees with \(n\) edges such that all trees with \(n\) edges are contained in such a graph whenever it contains the trees in the minimum set; second, all mutually nonisomorphic graphs that contain all trees with \(n\) edges.