On Potentially H-Graphic Sequences

For given a graph H, a graphic sequence Ï€ = (d₁, d₂,..., d_n) is said to be potentially H-graphic if there is a realization of Ï€ containing H as a subgraph. Let σ(K_r+1 — E₃,n) be the smallest even integer such that each n-term graphic sequence Ï€ = (d₁, d₂, ..., d_n) with term sum σ(Ï€) = d₁ + d₂ + ... + d_n ≥ σ(K_r+1 — E₃, n) has a realization containing K_r+1 — E₃ as a subgraph, where E₃ = {e₁, e₂, e₃} and K_r+1 — E₃ is a graph obtained from a complete graph K_r+1 by deleting three edges e₁,e₂,e₃ which form a triangle. In this paper, we characterize the potentially K₅ — e-graphic sequences without zero terms and give two simple sufficient and necessary conditions for a positive graphic sequence Ï€ to be potentially K₅-graphic, where K_r is a complete graph on r vertices and K_r — e is a graph obtained from K_r by deleting one edge. Moreover, we also give a simple sufficient and necessary conditions for a positive graphic sequence Ï€ to be potentially K₆-graphic. Finally, we determine the value σ(K_r+1 — E₃, n) for r ≥ 3 and n ≥ 3r + 5.