10 problems

Let $T$ be a tree. For a vertex $w$ and a neighbor $x$ of $w$, let $T^x$ denote the component containing $x$ after deleting $w$, let $\mu(x)$ denote the number of isomorphism types…

Primitive homogeneous structures conjecture. The distinguishing number of every primitive homogeneous countably infinite structure is two or infinite.

Let $S_n$ act on a set, and let its distinguishing number be the least number of labels needed to eliminate all nonidentity elements of the action. The large-degree distinguishing-…

Let $G$ be a connected graph of order $n\geq 3$, and let $\mu(G)$ denote its Mycielskian. Write $D(G)$ for the distinguishing number of $G$ and $D'(G)$ for its distinguishing index…

Regularity and component conjecture. (i) If $G$ is a $d$-distinguishing critical graph, then $G$ is a $k$-regular graph for some $k\leq d$. (ii) If $G$ is a disconnected $d$-distin…

Distinguishing stability conjecture.

Motion conjecture for graphs. The inequality

Infinite motion conjecture for graphs. If $\operatorname{Aut}(G)$ has infinite motion, then

Infinite motion conjecture for permutation groups. If $A$ is closed and subdegree-finite with infinite motion, then

Let $\Gamma$ be a connected, locally finite, denumerable graph. Its automorphism group has infinite motion if it contains no automorphism with finite support, where the support of…