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.

A graph \(H\) is \underline{collapsible} if for every even subset \(W \subseteq V(H)\), \(H\) has a spanning connected subgraph whose set of odd-degree vertices is \(W\). In a graph \(G\), there is a unique collection of maximal collapsible subgraphs, and when all of them are contracted, the resulting contraction of \(G\) is a reduced graph. Reduced graphs have been shown to be useful in the study of supereulerian graphs, hamiltonian line graphs, and double cycle covers, (see[2], [3], [4] [6] ), among others. It has been noted that subdividing an edge of a collapsible graph may result in a noncollapsible graph. In this note we characterize the reduced graphs of elementary subdivision of collapsible graphs of diameter at most two. We also obtain a converse of a result of Catlin [3] when restricted to graphs of diameter at most two. The main result is used to study some hamiltonian property of line graphs.

The \(F\)-free chromatic number \(\chi(M:-F)\) of a graph \(M\) is defined as the least number of classes in a partition of the vertices of \(M\) such that \(F\) does not occur as an induced subgraph in the subgraph induced by any of the colour classes. Two graphs \(G\) and \(H\) are called chromatically related if, for each positive integer \(k\), there exists a graph \(M\) such that \(\chi(M:-G) = \chi(M:-H) = k\), and distantly related whenever a chain of such relatednesses exists between them. Using a basic theorem of Folkman [3], we show that every two graphs on at least two vertices are distantly related.

BIBRC (balanced incomplete block with nested rows and columns) designs were introduced by Singh and Dey [1979] and these designs were mostly obtained by trial and error. Agrawal and Prasad [1983] gave some systematic methods of construction of these designs. We provide further systematic and general methods of construction of BIBRC designs in the present note.

An exponent bound is presented for abelian \((p^{i+j}, p^i, p^{i+j},p^j)\) relative difference sets: this bound can be met for \(i \leq j\).

A smallest transversal of a \(k\)-graph (or \(k\)-uniform hypergraph) is any smallest set of vertices that intersects all edges. We investigate smallest transversals of small (up to ten vertex) \(3\)-graphs. In particular, we show how large the smallest transversal of small \(3\)-graphs can be as a function of the number of edges and vertices. Also, we identify all \(3\)-graphs with up to nine vertices that have largest smallest transversals. This work is related to a problem of Turán, and to the covering problem. In particular, extremal \(3\)-graphs correspond to covering designs with blocks of size \(n-3\).