On the decomposition of a complete graph into planar subgraphs

Abstract

This paper considers the decomposition of a complete graph into planar subgraphs such that the union of all these planar subgraphs is the original complete graph, and no two of them have any edge in common. The motivation for this problem is the synthesis of a given electrical network with as few printed circuits as possible. The main result is that the smallest number of planar subgraphs into which the complete graph K n of n vertices can be decomposed does not exceed 〈 n 4〉 , where 〈x〉 stands for the minimum integer not less than real number x.