On vertex-transitive graphs of odd prime-power order

Marusic (Ann. Discrete Math. 27 (1985) 115) proved that all vertex-transitive graphs of order pk are Cayley graphs for each prime p and k = 1,2, or 3, and constructed a non-Cayley vertex-transitive graph of order pk and valency 2p + 2 for each prime p ≥ 5 and k ≥ 4. McKay and Praeger (J. Austral. Math. Soc. (A) 56 (1994) 53) gave an alternative construction of a non-Cayley vertex-transitive graph of order pk for each prime p ≥ 3 and k ≥ 4. In this paper it is proved that, for each positive integer k and each prime p ≥ 3, a vertex-transitive graph of order pk with valency less than 2p + 2 is a Cayley graph.

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