Given a graph H, a graphic sequence ? is potentially H-graphic if there is a
realization of ? containing H as a subgraph. In 1991, Erd?s et al.
introduced the following problem: determine the minimum even integer ?(H,n)
such that each n-term graphic sequence with sum at least ?(H,n) is
potentially H-graphic. This problem can be viewed as a ?potential? degree
sequence relaxation of the Tur?n problems. Let H be an arbitrary graph of
order k. Ferrara et al. [Combinatorica, 36(2016)687-702] established an
upper bound on ?(H,n): if ? = ?(n) is an increasing function that tends to
infinity with n, then there exists an N = N(?,H) such that ?(H,n)? ?~(H)n
+ ?(n) for any n ? N, where ?~(H) is a parameter only depending on the graph
H. Recently, Yin [European J. Combin., 85(2020)103061] obtained a new upper
bound on ?(H,n): there exists an M = M(k,?(H)) such that ?(H,n) ? ?~(H)n
+ k2-3k+4 for any n ? M. In this paper, we investigate the precise
behavior of ?(H,n) for arbitrary H with ?~?(H)+1(H < ?~(H) or??(H)+1(H) ?
2, where ??(H)+1(H) = min{?F)|F is an induced subgraph of H and |V(F)|=
?(H) + 1} and ?~?(H)+1(H) = 2(k-?(H)-1)+??(H)+1(H)-1. Moreover, we also
show that ?(H,n) = (k-?(H)-1)(2n-k+?(H))+2 for those H so that
??(H)+1(H) = 1,?~?(H)+1(H)=~?~(H),?~p(H) < ?~(H) for ?(H) + 2 ? p ? k and
there is an F < H with |V(F)| = ?(H) + 1 and ?(F) = (12,0?(H)-1).
Certain Bounds of Regularity of Elimination Ideals on Operations of Graphs