Subdividing a Graph Toward a Unit-distance Graph in the Plane

Abstract

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 K_nlies 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).