Chunhui Lai

A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determined the values of $\sigma (K_{r+1}-H, n)$ for $n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ and $H$ be a graph on $k$ vertices which containing a tree on $4$ vertices but not containing a cycle on $3$ vertices and $\sigma (K_{r+1}-P_2, n)$ for $n\geq 4r+8, r\geq 3$. There are a number of graphs on $k$ vertices which containing a tree on $4$ vertices but not containing a cycle on $3$ vertices. (For example, the cycle on $k$ vertices, the tree on $k$ vertices, and the complete 2-partite graph on $k$ vertices, etc. )