Let $G = (V,E)$ be a loop-free undirected graph, where $|V| = 6$ and $deg(v) = 2$ for all $v \in V$, Up to isomorphism how many such graphs $G$ are there?
I understand this question as asking how many non-isomorphic graphs there are that satisfy:
- $G$ is a loop-free undirected graph
- $|V| = 6$
- $deg(v) = 2$ for all $v \in V$
The solution given is 2: a cycle on six vertices and a disjoint union of two cycles (each on three vertices).
How come the answer isn't 3? The two solutions given above and also a disjoint union of three cycles (each on two vertices).
So something like 3 copies of the two-vertices cycle below.
Unless stated otherwise, a graph has at most one edge between any pair of vertices. Put another way, an edge is completely defined by its two endpoints, and if two edges have the same endpoints then they are in fact the same edge. Your suggestion of three disjoint 2-cycles doesn't work because 2-cycles are never permitted.
(When this condition is relaxed, the resulting object is usually called a multigraph.)
If 2-cycles were allowed, you would have missed a fourth solution: the multigraph could have been the union of a 2-cycle and a 4-cycle.