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.

For the notation used to express permutations, see Cycle decomposition (group theory).

Cycle decomposition of ${\displaystyle K_{n}}$ and ${\displaystyle K_{n}-I}$

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

References

1. Alspach, Brian (2001). "Cycle Decompositions of Kn and Kn−I". Journal of Combinatorial Theory, Series B. 81: 77–99. doi:10.1006/jctb.2000.1996.

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