We prove a conjecture of Saito that if a graph G with delta greater than or
equal to 3 has no cycle of length 1 (mod 3), then G has an induced subgrap
h which is isomorphic to the Petersen graph. The above result strengthened
the result by Dean et al. that every 2-connected graph with delta greater t
han or equal to 3 has a (1 mod 3)-cycle if G is not isomorphic to the Peter
sen graph. (C) 2001 Elsevier Science B.V. All rights reserved.