Alspach's conjecture

For complete graphs ${\displaystyle K_{n}}$ whose number ${\displaystyle n}$ of vertices is even, Alspach conjectured that it is always possible to decompose the graph into a perfect matching and a collection of cycles of prescribed lengths summing to ${\displaystyle {\tbinom {n}{2}}-{\tfrac {n}{2}}}$. In this case the matching eliminates the odd degree at each vertex, leaving a subgraph of even degree, and the remaining condition is again that the sum of the cycle lengths equals the number of edges to be covered. This variant of the conjecture was also proven by Bryant, Horsley, and Pettersson.

The Oberwolfach problem on decompositions of complete graphs into copies of a given 2-regular graph is related, but neither is a special case of the other. If ${\displaystyle G}$ is a 2-regular graph with ${\displaystyle n}$ vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for ${\displaystyle G}$ would also provide a decomposition of the complete graph into ${\displaystyle (n-1)/2}$ copies of each of the cycles of ${\displaystyle G}$. However, not every decomposition of ${\displaystyle K_{n}}$ into this many cycles of each size can be grouped into disjoint cycles that form copies of ${\displaystyle G}$, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have ${\displaystyle (n-1)/2}$ copies of each cycle.

• Alspach, B. (1981), "Problem 3", Research problems, Discrete Mathematics, 36 (3): 333, doi:10.1016/s0012-365x(81)80029-5
• Bryant, Darryn; Horsley, Daniel; Pettersson, William (2014), "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths", Proceedings of the London Mathematical Society, Third Series, 108 (5): 1153–1192, arXiv:1204.3709, doi:10.1112/plms/pdt051, MR 3214677
• Chartrand, Gary; Lesniak, Linda; Zhang, Ping (2015), "Alspach's conjecture", Graphs & Digraphs, Textbooks in Mathematics, vol. 39 (6th ed.), CRC Press, p. 349, ISBN 9781498735803