# Is this a sufficient condition for vertex and edge transitivity?

I'm traying to confirm (or refute) the adhering to statement:

Any linked $r$ - normal chart of $g$ such that every side is shared by the very same variety of minimum size cycles (that is, cycles of size $g$), is and also .

This is not a book workout. Any kind of suggestions valued.

Many thanks.

I think it is additionally incorrect for linked charts.

As an example, if you take the pentagonal prism (it is vertex transitive yet not side transitive) you can after that create a chart as adheres to: change each vertex by a triangular after that recognize vertices which are signed up with by a side.

This chart is 4 - normal and also of girth 3 and also every side remains in specifically one triangular.

Nonetheless it is not vertex or border transitive ; 10 vertices become part of 5 - cycles without chords yet 5 vertices aren't. Some sides hinge on a 4 - face and also some do not.

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