Abstract
Every simple graphG=(V, E) can be represented by a family of equal nonoverlapping spheres {S v :v ∈ V} in a Euclidean spaceR n in such a way thatuv ∈ E if and only ifS u andS v touch each other. The smallest dimensionn needed to representG in such a way is called the contact dimension ofG and it is denoted by cd(G). We prove that (1) cd(T)<(7.3) log |T| for every treeT, and (2)
whereK m +E n is the join of the complete graph of orderm and the empty graph of ordern. For the complete bipartite graphK n,n this implies (1.286)n−1 <cd(K n,n )<(1.5)n.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Alon, Explicit construction of exponeitial-sized families ofk-independent sets,Discrete Math. 58 (1986), 191–193.
P. Erdös and Z. Füredi, The greatest angle amongn points in thed-dimensional euclidean space,Ann. Discrete Math. 17 (1983), 275–283.
J. Friedman, ConstructingO(n logn) size monotone formula for thekth elementary symmetric polynomial ofn Boolean variables,Proceedings of the 25th IEEE Symposium on Foundations of Computer Science, 506–515, 1984.
H. Maehara, Contact patterns of equal nonoverlapping spheres,Graphs Combin. 1 (1985), 271–282.
R. A. Rankin, The closest packing of spherical caps inn-dimensions,Proc. Glasgow Math. Assoc. 2 (1955) 139–144.
J. Reiterman, V. Rödl, and E. Šiňajová, Geometrical embeddings of graphs,Discrete Math., to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Frankl, P., Maehara, H. On the contact dimensions of graphs. Discrete Comput Geom 3, 89–96 (1988). https://doi.org/10.1007/BF02187899
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187899