7 problems

Generalized Kriesell conjecture. For every $t>0$, every minimally $t$-tough graph has a vertex of degree $\lceil 2t\rceil$.

Co-diameter-two characterization conjecture. The graph $G$ is non-trivially minimally tough if and only if $G$ is isomorphic to $S_{\ell,\ell,\ell}$ for some $\ell\geq 1$.

Triply starred co-chordal conjecture. If the co-diameter of a co-chordal graph is $2$, then its non-trivially minimally tough graphs are exactly the graphs $S_{\ell,\ell,\ell}$ for…

Dallard et al.'s conjecture. There exists no minimally $t$-tough chordal graph for any real number $t>1/2$.

A graph $G$ is $t$-tough if $|S|\geq t c(G-S)$ for every subset $S\subseteq V(G)$ with $c(G-S)>1$, where $c(G)$ is the number of components of $G$. The 2-toughness conjecture. Ever…

Let $G$ be a graph with adjacency eigenvalues $\mu_1\ge \mu_2\ge\cdots\ge \mu_n$, minimum degree $\delta$, and toughness $t(G)$. Haemers's conjecture. Haemers conjectured that … A…

Let $G$ be a simple graph with minimum degree $\delta$. Let $\mu_2$ and $\mu_n$ denote the second-smallest and largest eigenvalues of the Laplacian matrix of $G$, respectively, and…