1 7 Se p 20 07 The smallest degree sum that yields potentially K r + 1 − Z - graphical Sequences ∗

Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graphH (H is a subgraph ofKm). We use the symbol Z4 to denote K4−P2. A sequence S is potentially Km−H-graphical if it has a realization containing aKm−H as a subgraph. Let σ(Km−H,n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − H,n) is potentially Km − H-graphical. In this paper, we determine the values of σ(Kr+1−Z, n) for n ≥ 5r+19, r+1 ≥ k ≥ 5, j ≥ 5 where Z is a graph on k vertices and j edges which contains a graph Z4 but not contains a cycle on 4 vertices. We also determine the values of σ(Kr+1 − Z4, n), σ(Kr+1 − (K4 − e), n), σ(Kr+1 −K4, n) for n ≥ 5r + 16, r ≥ 4.