The smallest degree sum that yields potentially kCℓ-graphic sequences

Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1, 1999, pp. 387–400) considered a variation of the classical Turán-type extremal problems as follows: For a given graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d1,d2,…,dn) with term sum σ(π)=d1+d2+⋯+dn⩾σ(H,n) has a realization G containing H as a subgraph. In this paper, for given integers k and ℓ, ℓ⩾7 and 3⩽k⩽ℓ, we completely determine the smallest even integer σ(kCℓ,n) such that each n-term graphic sequence π=(d1,d2,…,dn) with term sum σ(π)=d1+d2+⋯+dn⩾σ(kCℓ,n) has a realization G containing a cycle of length r for each r, k⩽r⩽ℓ.