Intersection Graphs for Families of Balls in Rn

1. INTRODUCTION If F is a finite family of sets, then the intersection graph r(F) is the graph with vertex-set F and edges the unordered pairs C, D of distinct elements of F such that C n D # 0. It is easy to see [6, p. 19] that every graph G is isomorphic to some intersection graph T(F). Some interesting classes of graphs have arisen by letting F range over families of balls in some metric space, such as arcs on a circle or intervals of the real line [4], or cubes, boxes or spherical balls in n-space [9, 11, 12]. For the case of balls in R" with the Euclidean norm, Guttman [5] and Havel [7] have defined the sphericity, sph(G), of a graph G to be the least dimension n in which G is isomorphic to F(F) for F some family of open (equivalently, closed) balls of radius 1 ; and Maehara [111 has defined the contact dimension, ed(G), to be the least n for which G is isomorphic to T(F) for F some family of closed balls of radius 1 such that no pair of balls intersects in more than one point. Maehara has shown that sph(G) < cd(G) < I V(G)I-I for all graphs G, and has studied sph(G) and cd(G) as functions of the structure of G [9-11]. Roberts [12] has defined the cubicity, cub(G), of G to be the least n for which G is isomorphic to F(F) for F some family of unit cubes with edges parallel to the Cartesian coordinate axes in R". Such cubes can be viewed as balls with respect to a different norm on R", and the question arises as to how the shape of the unit ball in an n-dimensional normed linear space is related to the least n in which G can be represented by an appropriate T(F). Havel [7] has shown that there are graphs of sphericity 2 but with arbitrarily large cubicity ; Fishburn [3] has shown that there are graphs G of cubicity 2 or 3 for which sph(G) > cub(G), but remarks that it is unknown whether sph(H) > cub(H) can hold for graphs H of arbitrarily large cubicity. In this paper, we are concerned with a different type of problem. Let F be the set of all graphs F(F), where F is a family of balls of arbitrary radii in R" …