The Non-Biplanar Character of the Complete 9-Graph

Abstract

Let us define a planar partition of a graph G as a pair {H1, H2} of subgraphs of G with the following properties(i)Each of H1 and H2 includes all the vertices of G.(ii)Each edge of G belongs to just one of H1 and H2.(iii)H1 and H2 are planar graphs.It is not required that H1 and H2 are connected. Moreover either of these graphs may have isolated vertices, incident with none of its edges.We describe a graph having a planar partition as biplanar.