Abstract
An infinite family of cubic edge- but not vertex-transitive graphs is constructed. The graphs are obtained as regular Z n -covers of K 3,3 where n= p 1 e 1 p 2 e 2 ⋯ p k e k where p i are distinct primes congruent to 1 modulo 3, and e i ⩾1. Moreover, it is proved that the Gray graph (of order 54) is the smallest cubic edge- but not vertex-transitive graph.
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