The maximum-independent-sets conjecture for k-chromatic ℓ-connected graphs with k at most 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 $3 k $ and $n 2$. If $G$ is an $n$-vertex $k$-chromatic

-connected graph, then

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

Theorem characterizes $G^*$ for large $n$, and this conjecture proposes that the same extremal graph works throughout the stated range.