The Characterization for a Graphic Sequence to have a Realization Containing K 1,1,s

A graphic sequence π?=?(d ₁, d ₂, . . . , d _n ) is said to be potentially K _1,1,s -graphic if there is a realization of π containing K _1,1,s as a subgraph, where K _1,1,s is the 1?× 1?× s complete 3-partite graph. In this paper, a simple characterization of potentially K _1,1,s -graphic sequences for s?≥ 2 and n?≥ 3s?+?1 is obtained. This characterization implies Lai’s conjecture on σ(K _1,1,s , n), which was confirmed by J.H. Yin, J.S. Li and W.Y. Li, and the values of σ(K _2,s , n) for s?≥ 4 and n?≥ 3s?+?1, where K _2,s is the 2?× s complete bipartite graph.