Cycle decomposition (graph theory)

Template:For

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 (a perfect matching) 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 $3 \leq m \leq n$ , 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 $m$ .