Cycles of length 1 modulo 3 in graph

We prove a conjecture of Saito that if a graph G with δ≥3 has no cycle of length 1 (mod 3), then G has an induced subgraph which is isomorphic to the Petersen graph. The above result strengthened the result by Dean et al. that every 2-connected graph with δ≥3 has a (1 mod 3)-cycle if G is not isomorphic to the Petersen graph.