Embeddings of graphs in euclidean spaces

Abstract

The dimension of a graph G=(V, E) is the minimum number d such that there exists a representation {Mathematical expression} and a threshold t such that xy εE iff {Mathematical expression}. We prove that d(G)≤n-x(G) and {Mathematical expression} where n=|V| and x(G) is the chromatic number of G; we present upper bounds for the dimension of graphs with a large girth and we show that the complement of a forest can be represented by unit vectors in R6. We prove that d(G)≥1/15 n for most graphs and that there are 3-regular graphs with d(G)≥c log n/log log n. © 1989 Springer-Verlag New York Inc.