# Subdivision Number of Large Complete Graphs and Large Complete Multipartite Graphs

• Severino V. Gervacio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

## 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(K m,n ) = (m − 1)(n − m) when nm(m–1). This result is used to show that the subdivision number of the complete graph is asymptotic to ($$n \above 0pt 2$$), its number of edges. Likewise, the subdivision number of the complete bipartite graph K m, 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.

## References

1. 1.
Gervacio, S.V., Maehara, H.: Subdividing a graph toward a unit distance graph in the plane. European Journal of Combinatorics 21, 223–229 (2000)