143 problems

Harary-polynomial comparability conjecture. Either $\chi_{\mathcal P}$ and $\chi_{\mathcal Q}$ are d.p.-equivalent or they are d.p.-incomparable.

Let $k\in\mathbb{N}$ and let $D_1,\dots,D_k$ be oriented graphs satisfying … Write $[D_1,\dots,D_k]$ for their join, and let $\operatorname{inv}$ denote inversion number. Dijoin co…

Vertex-deletion conjecture. There exists $v\in V(D)$ such that

Dijoin counterexample-range conjecture. For all $\ell,r\in\mathbb{N}$ with $\ell\geq 3$ or $r\geq 3$, there exist oriented graphs $L$ and $R$ such that

Dijoin conjecture.

Alon's inversion-number partition conjecture. For every two positive integers $k_1,k_2$, there exists an integer $f(k_1,k_2)$ such that every oriented graph $D$ with

Bang-Jensen et al.'s tightness conjecture. For every positive integer $n$, there exists an oriented graph $D$ such that

Belkhechine et al.'s inversion-number conjecture.

Let $G_1$ and $G_2$ be graphs with the same number of vertices, and let $\Delta$ be the dissimilarity measure obtained from the collections of horizontal homologies partitioned by t…

A graph's completion is the graph obtained by the completion operation referred to in the source, and $c_2$ denotes the $c_2$-invariant of a connected graph with at least three ver…

Asymptotic corank-growth conjecture. As $k\to\infty$,

Fixed-corank theta extremal conjecture. If $G\neq K_4$, then

Suppose that $\xi(G)$ is a 4-invariant of graphs extending the specialization of the $\mathfrak{sl}(2)$-weight system at $c=-\frac{3}{32}$. Let $G$ be a graph, let $x$ be a vertex…

Let $G$ be a perfect group and let $S$ be a generating set subject to the restrictions required for the square clustering coefficient. Write $\square(\operatorname{Cay}(G,S))$ for…

Let $G$ be a connected simple graph, and let $\mu^{*}(G)$ denote its saturation number, the minimum cardinality of a maximal matching. Let … be its harmonic index. A graph is subqu…

Let $T$ be a tree on $n$ vertices. For a graph $G$ with no isolated vertices, define its Laplacian ratio by … where $L(G)$ is the Laplacian matrix, $d(v)$ is the degree of $v$, and…

For integers $n$ and $m$, define … and let $E_{n,m}$ be the graph obtained from a clique $K_{n-d-1}$ and a path … by joining each clique vertex to both $v_d$ and $v_{d-1}$, and by…

Mostar index extremal graph conjecture. For any $n \ge 3$, the graph

Let $\mathcal{T}(n)$ denote the set of values of the spanning-tree count $\tau(G)$ over unrestricted graphs on $n$ vertices. Cayley's theorem gives $\tau(K_n)=n^{n-2}$, so the poss…

For each integer $n\geq 1$, let $\mathsf{S}_n(-1)=\{P_G(-1):G\text{ is a simple graph on }n\text{ vertices}\}$. Since $|P_G(-1)|$ counts acyclic orientations of $G$, the spectrum c…

Let a spider be a tree with at most one vertex of degree greater than two, and let $X_T^T$ denote the self-chromatic symmetric function of a spider $T$. Spider self-CSF conjecture.…

Let $\mathbf{G}$ be the set of isomorphism classes of finite simple graphs. For $G\in\mathbf{G}$, let $\varphi(G)$ be the graph invariant defined by … where $\chi_3(G)$ is the numb…

Let $E_n$ be the degree anti-regular graph, meaning the unique connected graph whose vertex degrees attain every value from $1$ through $n-1$. Let $\omega(E_n)$ denote its minimum…

Lower-bound conjecture for tournament dijoins. One should have

Alon–Powierski–Savery–Scott–Wilmer's n-dijoin conjecture. If