18 problems

Collins–Trenk conjecture. If $G$ is different from $K_{\Delta,\Delta}$ and $C_6$, then

Almost-all finite graphs conjecture. Almost all finite graphs are 2-distinguishable.

Let $F$ be a forest on $n$ vertices, and let $\chi'_{\cup}(F)$ denote its union vertex-distinguishing edge chromatic number, where edge colors are sets and the union of the colors…

Distinguishing-index conjecture. One has

Conjectured edge-distinguishing bound. Then

Conjectured infinite-motion bound. Then

Conjectured distinguishing-chromatic bound. Then

Let $G$ be a graph, and let $\mathcal{L}=\mathcal{L}$ be a collection of equal-sized lists, each of size $k$, indexed by the vertices $v eq 0$ of $G$. Choose a color from each vert…

Let $G$ be a connected simple graph. Let $\chi \, '_s(G)$ and $\chi \, '_{as}(G)$ denote the strong and adjacent-strong vertex-distinguishing proper edge chromatic numbers, respect…

Let $E(n)$ be the Euclidean isometry group acting on $\mathbf R^n$. Define the replacement number $R(\mathbf R^n,E(n))$ to be the smallest $m\in\mathbf N$ such that every distingui…

Let $E(2)$ be the Euclidean isometry group acting on $\mathbf R^2$, and let $\operatorname{ext}_D(\mathbf R^2,E(2))$ denote the distinguishing extension number. Finite extension-nu…

Let $E(2)$ be the Euclidean isometry group of $\mathbf R^2$, let $O(3)$ act on $S^2$, and let $E(3)$ be the Euclidean isometry group of $\mathbf R^3$. Their distinguishing extensio…

Let $E(3)$ denote the Euclidean isometry group acting on $\mathbf R^3$, and let $\operatorname{ext}_D(\mathbf R^3,E(3))$ be its distinguishing extension number. Finite extension-nu…

3-dimensional Euclidean extension conjecture.

2-sphere extension conjecture.

Plane extension conjecture.

Circle extension conjecture.

Cycle extension conjecture.