Graphs Having (2 mod d)-Cycles

It is known that if G is a graph with minimum degree δ at least d+1, then G has a cycle of length 2 mod d. We show that if G is also bipartite, then G has a cycle of length 2 mod 2d. Both these bounds are tight in terms of minimum degree. However, we show that if G is a graph with δ ≥ d and had neither Kd nor Kd,d as an induced subgraph, then G has a cycle of length 2 mod d. If G is also bipartite, then G has a cycle of length 2 mod 2d. If G is a 2-connected graph with δ ≥ d and is not congruent to Kd nor Kd,dʹ (for dʹ ≥ d) then G has a cycle of length 2 mod d. If G is also bipartite, then G has a cycle of length 2 mod 2d.