A sequence $S$ is potentially $K_{ p_{ 1},p_{ 2},...,p_{ t}}$ graphical if it has a realization containing a $K_{ p_{ 1},p_{ 2},...,p_{ t}}$ as a subgraph, where $K_{ p_{ 1},p_{ 2},...,p_{ t}}$ is a complete t-partite graph with partition sizes $p_{ 1},p_{ 2},...,p_{ t} (p_{ 1}\geq p_{ 2}\geq ...\geq p_{ t} \geq 1)$. Let $\sigma(K_{ p_{ 1},p_{ 2},...,p_{ t}}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{ p_{ 1},p_{ 2},...,p_{ t}}, n)$ is potentially $K_{ p_{ 1},p_{ 2},...,p_{ t}}$ graphical. In this paper, we prove that $\sigma (K_{ p_{ 1},p_{ 2},...,p_{ t}}, n)\geq 2[((2p_{ 1}+2p_{ 2}+...+2p_{ t}-p_{ 1}-p_{ 2}-...-p_{ i}-2)n -(p_{ 1}+p_{ 2}+...+p_{ t}-p_{ i})(p_{ i}+p_{ i+1}+...+p_{ t}-1)+2)/2]$ for $n \geq p_{ 1}+p_{ 2}+...+p_{ t}, i=2,3,...,t.$
Categories: math.CO