Abstract
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.
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Maehara, H. Contact patterns of equal nonoverlapping spheres. Graphs and Combinatorics 1, 271–282 (1985). https://doi.org/10.1007/BF02582952
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DOI: https://doi.org/10.1007/BF02582952