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 KnI

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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   and   with  , the graph   (where   is a 1-factor) can be decomposed into cycles of length   if and only if the number of edges in   is a multiple of  . Also, for positive odd integers   and   with  , the graph   can be decomposed into cycles of length   if and only if the number of edges in   is a multiple of  .

References

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  1. ^ Alspach, Brian (2001). "Cycle Decompositions of   and   ". 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.