Paintings: a planar approach to higher dimensions
An (n\(+1)\)-coloured graph is a finite and regular graph of degree \(n+1\) with an (n\(+1)\)-colouring of the edges such that at each vertex there is exactly one edge of each colour. PL closed n-manifolds can be encoded by certain such \((n+1)\)-coloured graphs called n-crystallizations. In this paper a related encoding by sets of n planar bipartite directed graphs, called n-paintings, is obtained. It is shown that a theorem of \textit{M. Ferri} and \textit{C. Gagliardi} [Pac. J. Math. 100, 85-103 (1982; Zbl 0517.57003)] giving an equivalence of crystallizations corresponding to homeomorphism of manifolds, can be stated in a rather simple way (restricted crystallization moves become deletion and insertion of edges in planar graphs). Properties of 3-paintings and minimum (i.e. least number of vertices encoding the same manifold) 3-crystallizations are obtained, and two conjectures stated.
scientific article; zbMATH DE number 4040858