Introducing Spacial Graphs ( updated

We introduce spatial graphs through a natural extension of the concept of pla-narity in three dimensions. For a graph G=(V,E), a face is a simple cycle of edges, and a complete set of faces is such that each edge belonging to a cycle in E is on at least one face in the set. G is spatial if admits a three-dimensional representation R with a complete set of faces consisting of simple surfaces, such that no two faces intersect except along the common edge. In particular G is called k-spatial if R includes k cells (spatial regions); and is called plane-face spatial if all the faces in R are plane. We prove that all graphs are 1-spatial, but not all of them are (k>1)-spatial or plane-face spatial. In particular, K n is (n 2 ? 3n)=2-spatial and plane-face spatial. We derive some basic properties of spatial graphs as natural three-dimensional extensions of the properties of planar graphs.