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I'm looking for graph $G$ such that $G$ is edge-transitive but $G^c$ is not edge-transitive.

My conjecture:If $G$ is edge-transitive then $G^c$ is edge-transitive

Please advise me.

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up vote 3 down vote accepted

An easy counterexample - the graph which is edge-transitive and its complement is not: simply take one edge in $K_4$ as you graph. (The complement is $K_4$ with one edge omitted.)

Graph:

line in K4

Complement:

Complement of line in K4

Complement can be redrawn like this:

Different plot of the same graph

Note that the edge joining the vertices of degree 3 cannot be mapped to any other edge.

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