If a 4 regular graph is hamiltonian can we say it is hamiltonian decomposable ?
Thanks a lot in advance
2026-03-07 19:29:50.1772911790
Hamiltonian decomposability of 4- regular graphs
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The MathWorld article on Hamiltonian decomposition gives the following examples of regular graphs which are Hamiltonian but not Hamiltonian decomposable:
All of these happen to be $4$-regular, which would probably be disappointing to someone who was looking for a different kind of example, but is exactly what you are looking for.
In the first four examples, it is easy to see why thy are not Hamiltonian decomposable: there are two parts connected by only two edges, and once a single Hamiltonian cycle is removed, they become disconnected. The last two examples don't have this property, so they are not Hamiltonian decomposeable for some other, subtler reason.