An Erdös-Stone Type Conjecture for Graphic Sequences

We consider a variation of the classical Turán-type extremal problem as introduced by Erdős, Jacobson and Lehel in [4]. Let π be an n-element graphic sequence and let H be a graph. We wish to determine the smallest even integer m such that any n-term graphic sequence π having degree sum at least m has some realization containing H as a subgraph. Denote this value m by σ(H,n). For an arbitrarily chosen H, we construct a graphic sequence π(H,n) whose degree sum plus two is at least σ(H,n). Furthermore, we conjecture that equality holds in general, as this is the case for all choices of H where σ(H,n) is currently known.