Subdivision number of large complete graphs and large complete multipartite graphs

Abstract

A graph whose vertices can be represented by distinct points in the plane such that points representing adjacent vertices are 1 unit apart is called a unit-distance graph. Not all graphs are unit distance graphs. However, if every edge of a graph is subdivided by inserting a new vertex, then the resulting graph is a unit-distance graph. The minimum number of new vertices to be inserted in the edges of a graph G to obtain a unit-distance graph is called the subdivision number of G, denoted by sd (G). We show here in a different and easier way the known result sd(Km,n) = (m - 1)(n - m) when n ≥ m(m - 1). This result is used to show that the subdivision number of the complete graph is asymptotic to (2n), its number of edges. Likewise, the subdivision number of the complete bipartite graph Km,n is asymptotic to mn, its number of edges. More generally, the subdivision number of the complete n-partite graph is asymptotic to its number of edges. © Springer-Verlag Berlin Heidelberg 2005.