11 problems

Minimal proper-interval completion conjecture. The only-if condition of that theorem is also sufficient. Moreover, if this condition is sufficient, then the problem of finding a mi…

Rzk{a}.{z}ewski's conjecture. Locally interval graphs are exactly circular-arc graphs.

Let $G$ be an interval graph. A graph is called BDM-constructible if it is uniquely determined by its boundary distance matrix (with the relevant order and boundary fixed). Interva…

Let $G$ be a finite simple undirected graph. For each vertex $v\in V(G)$, let $G_v=[G-v]_{\cong}$ be its card, and let … be its deck. The graph $G$ is reconstructible if every grap…

Let $G=([n],E)$ be an interval graph and let $\tau\in\mathfrak{S}_n$. Let $\Phi_G(\tau)$, $\mathcal{Q}_{\Phi_G(\tau)}^{(G)}$, $\mathsf{CoInv}_G$, and $\mathsf{Inv}_G$ be the functi…

Let $G=([n],E)$ be an interval graph. For a composition $\alpha\vDash n$, let $\Psi_\alpha$ be the type 1 power sum quasisymmetric function, let $z_\alpha=z_{\lambda(\alpha)}$, and…

Impropriety-spectrum conjecture. The cardinality $\left|\mathrm{spec}_{\mathrm{imp}}(G)\right|$ is unbounded, but is rather small in comparison to $p$; the precise size remains an…

Conjecture. $G$ is an interval $k$-graph if and only if it has no induced subgraph isomorphic to $\overline{C}_6$ or $\overline{2P}_3$.

Let $n\geq 1$ and let $v_1,\dots,v_n$ be distinct speeds. For $x\in\mathbb{R}$, write $\|x\|$ for the distance from $x$ to the nearest integer. Weak Lonely Runner Conjecture. There…

One-third approval-ratio conjecture. For all pairwise-intersecting double-interval societies $S$, the approval ratio

An interval graph is the intersection graph of a family of non-empty convex subsets of a linear order. For a class of graphs, let $\mathcal{C}_{\chi}$ mean that, for every graph…