We present an edge labeling of order-k Voronoi diagrams, V k (S), of point sets S in the plane, and study properties of the regions defined by them. Among them, we show that V k (S) has a small orientable cycle and path double cover, and we identify configurations that cannot appear in V k (S) for small values of k. This paper also contains a systematic study of well-known and new properties of V k (S), all whose proofs only rely on elementary geometric arguments in the plane. The maybe most comprehensive study of structural properties of V k (S) was done by D.T. Lee (On k-nearest neighbor Voronoi diagrams in the plane) in 1982. Our work reviews and extends the list of properties of higher order Voronoi diagrams.

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