Let $n$ be an odd positive integer. There is a graph $G$ with $2n$ vertices such that if you partition the vertices into two groups $A$ and $B$ with $n$ vertices each, then the subgraph consisting of only vertices and edges within $A$ is connected and has a closed path containing all of its edges, starting and ending with the same vertex. The same condition is true for $B$ as well. Is $G$ necessarily a clique? (Proposed by Ho Janson)