A NOTE ON GRAPHS AND SPHERE ORDERS

A partially ordered set P is called a k-sphere order if one can assign to each element a is-element-of P a ball B(a) in R(k) so that a < b iff B(a) subset-of B(b). To a graph G = (V, E) associate a poset P(G) whose elements are the vertices and edges of G. We have v < e in P(G) exactly when v is-an-element-of V, e is-an-element-of E, and v is an end point of e. We show that P(G) is a 3-sphere order for any graph G. It follows from E. R. Scheinerman [''A Note on Planar Graphs and Circle Orders, '' SIAM Journal of Discrete Mathematics, Vol. 4 (1991), pp. 448-451] that the least k for which G embeds in R(k) equals the least k for which P(G) is a k-sphere order. For a simplicial complex Kone can define P(K) by analogy to P(G) (namely, the face containment order). We prove that for each 2-dimensional simplicial complex K, there exists a k so that P(K) is a k-sphere order.