Determine piercing by cycles in the subquadratic range

Determine the asymptotics of g(n,Kp|Cq) for p≤2q−2, including the cases (p,q)=(5,4) and (p,q)=(6,4).

Background

Theorem 9.37 provides bounds in part of the range, but not tight asymptotics. The paper specifically identifies the first unresolved parameter pairs.