Counterexample to the Bougard-Joret Conjecture
For admissible integers $n,\alpha,k$, let $f(n,\alpha,k)$ be the minimum number of edges in a $k$-connected graph of order $n$ and independence number $\alpha$. A conjecture of Bougard and Joret predicts that $f(n,\alpha,k)=\lceil nk/2\rceil$ when $n\leq k\alpha$, under the assumptions $n\geq2\alpha$, $n\geq\alpha+k$, $\alpha\geq2$, and $k\geq3$. We disprove this prediction, determine $f(n,\alpha,k)$ throughout the boundary $n=\alpha+k$, and characterize every extremal graph on that boundary. In particular, for every $k\geq4$, \[ f(2k-1,k-1,k)=k^2-1, \] whereas the conjectured value is $k^2-\lfloor k/2\rfloor$. The extremal graphs in this family are precisely $\overline K_{k-1}\join T$, where $T$ is an arbitrary tree of order $k$. The smallest-order failure has parameters $(n,\alpha,k)=(7,3,4)$, and no admissible counterexample has smaller order.