The Erdős-Gyárfás conjecture (EGC) states that every graph with minimum vertex degree of at least 3 contains a cycle whose length is a power of 2. The statement has not been proven even for cubic graphs. Moreover, it has not been proven for cubic Hamiltonian graphs. On the other hand, we can see that every cubic Hamiltonian graph has an even number of vertices 2k and can be obtained by adding k edges to a 2k-cycle. It will be shown that adding this number of edges to a given vertex of the cycle always gives a 4-cycle. This suggests the validity of EGC for cubic Hamiltonian graphs. This theorem is a consequence of a more general result the proof of which is the main content of the paper. Namely, we will determine the maximum number of edges the addition of which to a given vertex of a cycle does not give a cycle of a sufficiently small length.