# When are the cardinalities of 2-factors in a graph equal?

Given a graph $G$, if we can partition the edges into pairwise disjoint subsets of $G$, such that the union of all the subsets is equal to the edgeset of G, then this is a decomposition. If such a partition can be formed from only spanning k-regular graphs, where all the graphs are disjoint, then each of these graphs is a k-factor, and the graph is k-factorizable. If a graph is 2-factorizable, then what conditions are nessasary to make the connected components of each 2-factor the same length?

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• All the factors are of equal size by definition. Do you mean to ask when the components of the factors are of equal size? – Brendan McKay Mar 5 '16 at 9:58
• I may just not understand what a factor is, then. Why are they of equal size? I suspect the question you proposed is what I meant to ask, and I just don't know it. – Juan Sebastian Lozano Mar 5 '16 at 10:02
• A 2-factor is a spanning 2-regular subgraph. So if the whole graph has $n$ vertices, a 2-factor has $n$ vertices and $n$ edges. Perhaps you are just trying to decompose the edge set into cycles? – Brendan McKay Mar 5 '16 at 10:08
• You are correct, and this question makes no sense in light of that. However, what is the difference between a two factor and a two factorization? But yes, that is exactly what I am trying to do, I am looking for cycles which are of equal length in the graph. – Juan Sebastian Lozano Mar 5 '16 at 10:12
• A 2-factorization is a partition of the edge set into 2-factors. Go to Google Scholar and search for "graph cycle decomposition". There are lots of partial results but no complete answer. You should do such searches before asking a question here. – Brendan McKay Mar 5 '16 at 10:16