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