Subdividing a graph toward a unit-distance graph in the plane

The subdivision number of a graph G is defined to be the minimum number of extra vertices inserted into the edges of G to make it isomorphic to a unit-distance graph in the plane. Let t (n) denote the maximum number of edges of a C-4-free graph on n vertices. It is proved that the subdivision number of K-n lies between n(n - 1)/2 - t(n) and (n - 2)(n - 3)/2 + 2, and that of K(m, n) equals (m - 1)(n - m) for n greater than or equal to m(m - 1). (C) 2000 Academic Press.