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We say that a connected, simple, undirected graph $G=(V,E)$ is edge-swapping if for every $e\in E$ there is a graph isomorphism $\varphi:G\to G$ such that for the restriction $\varphi|_e$ we have $\text{im}(\varphi|_e) = e$ and $\varphi|_e$ is not the identity (or, equivalently in this case, has no fixed points).

Is every edge-swapping graph edge-transitive?

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No - consider the vertex-and-edge graph of a truncated cube. Some but not all edges are part of 3-cycles, so this graph is not edge-transitive, but it is clearly edge-swapping.

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