Circumference of a graph and its distance dominating longest cycles

In this note, we prove the following: Let G be a k-connected graph (k≥2) with circumference c(G) and m a non-negative integer. Then • [(1)] Either c(G)≥(2m+2)k, or d_G(v,C)≤m for any longest cycle C and any vertex v of G. • [(2)] Either c(G)≥(2m+3)k, or d_G(e,C)≤m for any longest cycle C and any edge e of G. When m=0, C in (1) and (2) are well-known Hamiltonian cycle and dominating longest cycle, respectively. Moreover, we give graphs to show that the bounds on c(G) are all sharp, even for those graphs that are triangle-free with only possible exception m=0 in (2). (1) is also best possible for those graphs that are bipartite.