Potentially Kr1,r2,…,rl,r,sKr1,r2,…,rl,r,s-graphic sequences

A variation of a classical Turán-type extremal problem is considered as follows: determine the smallest even integer σ(Kr1,r2,…,rℓ,r,s,n)σ(Kr1,r2,…,rℓ,r,s,n) such that every n  -term graphic sequence π=(d1,d2,…,dn)π=(d1,d2,…,dn) with term sum σ(π)=d1+d2+⋯+dn⩾σ(Kr1,r2,…,rℓ,r,s,n)σ(π)=d1+d2+⋯+dn⩾σ(Kr1,r2,…,rℓ,r,s,n) has a realization G   containing Kr1,r2,…,rℓ,r,sKr1,r2,…,rℓ,r,s as a subgraph. In this paper, we determine σ(Kr1,r2,…,rℓ,r,s,n)σ(Kr1,r2,…,rℓ,r,s,n) for sufficiently large n  , where s⩾r⩾rℓ⩾⋯⩾r1⩾0s⩾r⩾rℓ⩾⋯⩾r1⩾0 and r⩾3r⩾3.