I have a question regarding an $n$-regular (simple) graph, and why it cannot be partitioned into paths of length $n+2$, for example.
For instance, if I have a $7$-regular simple graph, then I assume I have an $8$ vertex graph (since an $n$-vertex graph is $(n-1)$-regular). Then, if I needed to partition into paths of at least $9$ edges, that is impossible because a path cannot repeat vertices more than once (and we only have $8$, not $9$). Is my reasoning correct?