Are graphs with certain degree sequences complete? Let $G$ be a graph of order $n$. Let $d_1,d_2,....,d_n$ be the degree sequence of the graph (The degrees are arranged in a descending order, therefore $d_1\geq d_2\geq ...\geq d_n$. Suppose that $d_i+i\geq n$ for all $1\leq i\leq n$. Does it follow that $G$ is complete ?
Thank you  

Not necessarily. Using your own example, the red edges in the picture below show a graph with degree sequence $4,4,3,3,2$.

