5 problems

Let $G$ be a simple graph, let $v\in V(G)$, and let $G-v$ denote the graph obtained by deleting $v$. Twin-pair conjecture. If … then $G$ has at least one pair of twin vertices. Thi…

Let $C$ be a cograph, and let $\widetilde{T}$ be the tree obtained from its cotree by erasing the leaves. Cograph cotree lower-bound conjecture. … The proposed bound would sharpen…

Let $\mathcal{G}=\{G_i\}_{i=1}^{\infty}$ be an infinite family of simple graphs such that $G_i$ is a proper induced subgraph of $G_j$ for all $i<j$. Eventual family conjecture. Eit…

For $k\geq1$, let … and let $\mathcal{T}_{\delta}^*(G)$ denote the set of induced subgraphs of one graph in $\mathcal{T}_{\delta}(G)$. Finite classification conjecture. For every…

Let $G$ be a simple graph, let $delta":[0,1,-1]^{V(G)}$ be a vector, and let $\mathcal{T}_{\delta}(G)$ denote the associated family of graphs. A graph is minimal $k$-forbidden if i…