In a paper of Leighton, he conjectures that every vertex transitive graph has a multicycle decomposition. A multicycle decomposition is an edge colouring such that each colour class gives a spanning subgraph which is a disjoint union of cycles of the same length. This was proven to be false by Marušič, the counter example being the line graph of the Petersen graph. Does a weaker statement hold, where we remove the condition that the cycles have to be the same length?
I know there exists a lot of results about factorisations of graphs. In this language an in-between result would be 'all vertex transitive graph are 1/2- factorable (where we allow a mix).' I was wondering if this result has already proven? Otherwise if there where partial similar results? For example, cubic vertex transtive graph have a perfect matching, so we get an affirmative answer here.
In finite degree graphs there is a 2-factorisation of even degree regular graphs presented by Petersen that uses Hall's marriage theorem. Therefore, if an odd degree regular graph has a perfect matching, using the Petersen result we get the 1/2-factorisation we are looking for.
For vertex-transitive graphs, it is shown by Godsil and Royal (Algebraic Graph Theory, Theorem 3.5.1) that a finite vertex-transitive graph have such a perfect matching. This can be extended to countably infinite graphs, this will be done in a paper by Agelos Georgokopolous, Mattias Hamann, and myself. Along with the formalisation of what is above.