17 problems

Sillke's rank conjecture. The rank of $A(G)$ is equal to the number of distinct non-zero rows of $A(G)$.

Let $G$) be a connected cograph with no universal vertex. For the co-components $H_{(1)}$ and $H_{(2)}$ defined as in Proposition, Theo-Conjecture. … The cograph condition is essen…

Power-to-enhanced-power cograph conjecture. For every finite group $G$, if the power graph $\mathcal{P}(G)$ is a cograph, then the enhanced power graph $\mathcal{E}(G)$ is a cograp…

Let $H$ be a graph, let $S$ be a finite set of graphs, and write $\operatorname{Forb}_\subseteq(S)$ for the graphs containing no member of $S$ as an induced subgraph. One-forbidden…

Discrete–continuum exponent conjecture. With probability tending to $1$ as $n\to\infty$,

Eight-vertex bound. For all $F\in\mathfrak{F}_{\mathrm{GT}}$, we have

Cograph formulation of the Erdős–Hajnal conjecture. For every graph $H$, there exists an $\varepsilon>0$ such that every $n$-vertex $H$-free graph $G$ contains an induced cograph o…

Abrishami's conjecture. $G$ and $H$ have at least

Recursive characterization problem. Find a recursive characterization for the cograph minimal $(\infty,k)$-polar obstructions.

Uniqueness conjecture. There exists exactly one cograph minimal $(\infty,k)$-polar obstruction of type $(c,i)$. This extends the cases for which uniqueness is established and leave…

Ghaddar and Hliněný's multiplicity conjecture. For every eigenvalue $\lambda\ne0,-1$ of $G$, its multiplicity satisfies

Path conjecture. The minimum value of $M_G$ among all connected graphs of order $n$ is attained by the path $P_n$.

Let $G$ be a $P_4$-free graph, also called a cograph. The edge clique cover number $\theta_\mathrm{e}(G)$ is the minimum cardinality of a family of cliques such that every edge of…

Let $G$ be a cograph, and let $G/\!\equiv$ denote the quotient by the vertex equivalence relation introduced in the paper. Partially order the equivalence classes by the relation…

Let $G$ be a cograph. Introduce an equivalence relation on its vertices by identifying vertices with the same relevant open or closed neighborhoods, and partially order the resulti…

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…

A cograph is a graph obtained from a single vertex by repeatedly applying disjoint union and complementation. For a graph $G$, the edge-clique cover problem asks whether the edges…