4 problems

Clique-rank bound conjecture. For every positive integer $\ell$, if $G \in \mathcal{K}_{\ell+2,\ell-1}$ and $\omega(G) \geq 4$, then

Maximal-gamma conjecture. For every positive integer $\ell$, the maximum value of $\gamma_{\ell-1}(G,a)$ among $\ell$-minimal graphs is attained by a sparse stretched clique…

Sparsest minimal-graph conjecture. For every positive integer $\ell$, a sparsest $\ell$-minimal graph is a sparse stretched clique in $\hat{\mathcal{K}}_{\ell+2,\ell-1}$.

Enumeration conjecture. There are exactly $49$ non-isomorphic $3$-minimal graphs.