Does anyone have any examples of NP-hard graph problems which stay NP-hard on cycles, or is this class somehow not able to have NP-hard problems?
I found a similar post concerning trees here which answers affirmatively, but by using the fact that any $n$-size input can be encoded as a tree of order $n$. Cycles only have one member per order and thus this trick cannot be used. Due to this, I've started to think that the only parameter of a cycle graph is just the order $n$, and so perhaps graph problems restricted on cycle graphs don't really make sense as it's more of a problem on integers? However, since there are problems on integers which could be NP-hard, such as integer factorization, could it be that NP-hard problems on cycles do exist?
I checked on graphclasses but no ``cycle'' graph class exists, which further confirms my suspicion that there's just something inherently too simple about this class.