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 Kn and Kn − 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. 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 $$m$$.