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We introduce a representation via (n+1)-colored graphs of compact n-manifoldswith (possibly empty) boundary, which appears to be very convenient forcomputer aided study and tabulation. Our construction is ageneralization toarbitrary dimension of the one recently given by Cristofori and Mulazzani indimension three, and it is dual to the one given by Pezzana in the seventies.In this context we establish some results concerning the topology of therepresented manifolds: suspension, fundamental groups, connected sums and movesbetween graphs representing the same manifold. Classification results ofcompact orientable 4-manifolds representable by graphs up to six vertices areobtained, together with some properties of the G-degree of 5-colored graphsrelating this approach to tensor models theory.

Compact n-manifolds via (n+1)-colored graphs: a new approach