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A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least 2 (n−54)/4 nonisomorphic minimal… (More)

This is a preprint of an article accepted for publication in Acta Mathe-matica Universitatis Comenianae c 2004 (copyright owner as specified in the journal). Abstract It is shown that each possible pair of the 80 isomorphism classes of Steiner triple systems of order 15 may be realized as the colour classes of a face 2-colourable triangulation of the… (More)

Given two triangular embeddings f and f ′ of a complete graph K and given a bijection φ : V (K) → V (K), denote by M (φ) the set of faces (x, y, z) of f such that (φ(x), φ(y), φ(z)) is not a face of f ′. The distance between f and f ′ is the minimal value of |M (φ)| as φ ranges over all bijections between the vertices of K. We consider the minimal nonzero… (More)