The maximum-independent-sets conjecture for k-chromatic ℓ-connected graphs with k greater than ℓ

Let $G^*$ denote the graph characterized in the paper as the extremal $n$-vertex $k$-chromatic

-connected graph, and let $i(G)$ be the number of independent sets in $G$. **The maximum-independent-sets conjecture.** Let $2 < k$ and $n5$. If $G$ is an $n$-vertex $k$-chromatic

-connected graph, then

$$i(G) i(G^*).$$

The conjecture is known for $k=3$ and $=2$ when $n5$, while the general case remains open.