Packing Circuits into KN

Abstract

We shall pack circuits of arbitrary lengths into the complete graph KN. More precisely, if N is odd and [sum ]ti=1mi = (N2), mi [ges ] 3, then the edges of KN can be written as an edge-disjoint union of circuits of lengths m1,…,mt. Since the degrees of the vertices in any such packing must be even, this result cannot hold for even N. For N even, we prove that if [sum ]ti=1mi [les ] (N2) − N−2 then we can write some subgraph of KN as an edge-disjoint union of circuits of lengths m1,…,mt. In particular, KN minus a 1-factor can be written as a union of such circuits when [sum ]ti=1mi = (N2) − N−2. We shall also show that these results are best possible.