Sphericity, cubicity, and edge clique covers of graphs

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd. The edge clique cover number θ(G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G) ≤θ(G). Our upper bound remains valid for intersection graphs defined by balls in the Lp-norm for 1 ≤ p ≤ ∞