Cycle decomposition (graph theory)

In graph theory, a cycle decomposition is a decomposition (a partitioning of a graph's edges) into cycles. Every vertex in a graph that has a cycle decomposition must have even degree.

Cycle decomposition of $K_n$ and $K_n-I$

Brian Alspach and Heather Gavlas established necessary and sufficient conditions for the existence of a decomposition of a complete graph of even order minus a 1-factor into even cycles and a complete graph of odd order into odd cycles.^[1] Their proof relies on Cayley graphs, in particular, circulant graphs, and many of their decompositions come from the action of a permutation on a fixed subgraph.

They proved that for positive even integers $m$ and $n$ with $4\leq m\leq n$,the graph $K_n-I$ (where $I$ is a 1-factor) can be decomposed into cycles of length $m$ if and only if the number of edges in $K_n-I$ is a multiple of $m$. Also, for positive odd integers $m$ and $n$ with 3≤m≤n, the graph $K_n$ can be decomposed into cycles of length $m$ if and only if the number of edges in $K_n$ is a multiple of $m$.

References

1. Science Direct Article

• Bondy, J.A.; Murty, U.S.R. (2008), "2.4 Decompositions and coverings", Graph Theory, Springer, ISBN 978-1-84628-969-9.