Contact patterns of equal nonoverlapping spheres

LetF be a set of nonoverlapping spheres in Euclideann-spaceE ⁿ . By the contact pattern ofF we mean the graph whose vertex set isF and two vertices are adjacent whenever the corresponding spheres touch each other. Every graph turns out to be a contact pattern in some dimension. This paper studies the smallest dimensionn for a graphG such thatG is a contact pattern inE ⁿ . Among others, the smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.