Existence of graphs with specified cycle lengths (Q913817)

The two following theorems are proved. Let n,g\(\geq 3\) and let \(I\subseteq \{3,...,g\}.\) There exists an n-regular n-connected graph G such that for every \(i\in \{3,...,g\},\) G has a cycle of length i if and only if \(i\in I\). Let \(m,d\geq 1\) and let \(J\subseteq \{0,1,...,d\}.\) There exists an m- connected graph H such that for every \(i\in \{0,1,...,d\},\) H has a cycle of length v(H)-i if and only if \(i\in J\).