Alspach's conjecture
Alspach's conjecture is a mathematical theorem that characterizes the disjoint cycle covers of complete graphs with prescribed cycle lengths. It is named after Brian Alspach, who posed it as a research problem in 1981. A proof was published by . Formulation. In this context, a disjoint cycle cover is a set of simple cycles, no two of which use the same edge, that include all of the edges of a graph. For a disjoint cycle cover to exist, it is necessary for every vertex to have even degree, because the degree of each vertex is two times  the number of cycles that include that vertex, an even number. And for the cycles in a disjoint cycle cover to have a given collection of lengths, it is also necessary for the sum of the given cycle lengths to equal the total number of edges in the given graph. Alspach conjectured that, for complete graphs, these two necessary conditions are also sufficient: if formula_1 is odd (so that the degrees are even) and a given list of cycle lengths (all at most formula_1) adds to formula_3 (the number of edges in the complete graph) then the complete graph formula_4  can always be decomposed into cycles of the given length. It is this statement that Bryant, Horsley, and Pettersson proved. Generalization to even numbers of vertices. For complete graphs formula_4 whose number formula_1 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 formula_7. 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. Related problems. 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 formula_8 is a 2-regular graph, with formula_1 vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for formula_8 would also provide a decomposition of the complete graph into formula_11 copies of each of the cycles of formula_8. However, not every decomposition of formula_4 into this  many cycles of each size can be grouped into disjoint cycles that form copies of formula_8, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have formula_11 copies of each cycle.