On embedding triangle-free graphs in unit spheres

The study of tiispersed arrangements of points in certain spaces has attracted many authors. lit has produced some intricate constructions and interesting problems. Far a survey of some aspects of this study see 15). In this paper, the study of dispersed arrangements of points has ted to the following conjecture 121: If G is ti triangle-free graph, then there is a function cp : G --+ S’, where Sk denotes the unit sphere in the Euclidean space EL+‘, such that (g, g’)E E(G) if and only if flp(g)- +‘)I] 7> \/3. In this paper. we establish the conjecture for certain families of graphs. in particular, we show that bipartite graphs are embeddable in Sk and obtain bounds on the dimension of the sphere in which a bipartite graph can be embedded.