On vertex-transitive graphs of odd prime-power order

Marusic (Ann. Discrete Math. 27 (1985) 115) proved that all vertex-transitive graphs of order p(k) are Cayley graphs for each prime p and k = 1, 2, or 3, and constructed a non-Cayley vertex-transitive graph of order p(k) and valency 2p + 2 for each prime p greater than or equal to 5 and k greater than or equal to 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 p(k) for each prime p greater than or equal to, 3 and k greater than or equal to 4. In this paper it is proved that, for each positive integer k and each prime p greater than or equal to 3, a vertex-transitive graph of order p(k) with valency less than 2p + 2 is a Cayley graph. (C) 2002 Elsevier Science B.V. All rights reserved.