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. Lett (n) denote the maximum number of edges of a C4-free graph on n vertices. It is proved that the subdivision number of Knlies betweenn (n − 1) / 2 − t(n) and (n − 2)(n − 3) / 2 + 2, and that of K(m, n) equals (m − 1)(n − m) forn ≥ m(m − 1).