Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π = (d1, d2, ..., dn) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realize π. For a given graph H, a graphic sequence π is potentially H-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n ∈ Z+, σ(H, n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π) ≥ m is potentially H-graphic. Let K s denote the complete t partite graph such that each partite set has exactly s vertices. We show that σ(K s, n) = σ(K(t−2)s +Ks,s, n) and obtain the exact value of σ(Kj +Ks,s, n) for n sufficiently large. Consequently, we obtain the exact value of σ(K s, n) for n sufficiently large.