Potentially Kr+1-p-graphic sequences

Let 0 <= p <= [r+1/2] and sigma(K(r+1)(-p), n) be the smallest even 2 integer such that each n-term graphic sequence with term sum at least sigma(K(r+1)(-p), n) has a realization containing K(r+1)(-p) as a subgraph, where K(r+1)(-p) is a graph obtained from a complete graph K(r+1) on r + 1 vertices by deleting p edges which form a matching. In this paper, we determine sigma(K(r+1)(-p), n) for r >= 2, 1 <= p <= [r+1/2] and n >= 3r + 3. As a corollary, we also determine sigma(K(1n), (2t), n) for t >= 1 and n >= 3s + 6t, where K(1n), (2t) is an r(1) x r(2) x ... x r(s+t) complete (s + t)-partite graph with r(1) = r(2) = ... = r(s) = 1 and r(s+1) = r(s+2) = ... = r(s+t) = 2 and sigma(K(1n), (2t), n) is the smallest even integer such that each n-term graphic sequence with term sum at least sigma(K(1n), (2t), n) has a realization containing K(r+1)(-p) as a subgraph.