Take the 2-minute tour ×
Theoretical Computer Science Stack Exchange is a question and answer site for theoretical computer scientists and researchers in related fields. It's 100% free, no registration required.

Let $G=(V,E)$ be an undirected graph. A decomposition of $V$ into disjoint subsets $V_i$ is called a Hamilton decomposition of $G$ if the subgraph induced by each set $V_i$ is either a Hamilton graph or consists of a single edge with $|V_i|=2$.

Example: The complete bipartite graph $K_{m,n}$ possesses a Hamilton decomposition if and only if $m=n$.

I am looking for an algorithm that decides whether a given graph possesses a Hamilton decomposition. Is this decision problem NP-complete? If not, how can we find such a decomposition?

Note: In the literature a Hamilton decomposition often denotes a decomposition of the edges $E$ of $G$ such that the induced subgraphs are Hamilton. In contrast I am interested in a decomposition of the vertices.

share|improve this question

1 Answer 1

up vote 17 down vote accepted

If we request that each $|V_i| \ge 3$, then this is the 2-factor problem, see the book Combinatorial Optimization by Schrijver. If you allow $|V_i| = 2$, then we can solve this by replacing each undirected edge by two directed ones and compute what is called a cycle-cover. This can be done in polynomial time by reduction to bipartite matching.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.