On the contact dimensions of graphs

Abstract

Every simple graph G=(V, E) can be represented by a family of equal nonoverlapping spheres {Sv:v ∈ V} in a Euclidean space Rn in such a way that uv ∈ E if and only if Su and Sv touch each other. The smallest dimension n needed to represent G in such a way is called the contact dimension of G and it is denoted by cd(G). We prove that (1) cd(T)