On potentially Kr+1 - U-graphical sequences

Abstract

Let K_m - H be the graph obtained from K_m by removing the edges set E(H) of the graph H (H is a subgraph of K_m). We use the symbol Z₄ to denote K₄ - P₂. A sequence S is potentially K_m - H-graphical if it has a realization containing a K_m-H as a subgraph. Let σ(K_m-H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(K_m - H,n) is potentially K_m - H-graphical. In this paper, we determine the values of σ(K_r+1 - U, n) for n ≥ 5r +18, r +1 ≥ k ≥ 7,j≥6 where U is a graph on A; vertices and j edges which contains a graph K₃ ∪ P₃ but not contains a cycle on 4 vertices and not contains Z₄.