12 problems

Let $G$ be a graph and let $x$ be a vertex that is not a cut vertex of $G$. Write ${\rm gp}_{\rm o}(G)$ for the outer general position number of $G$ and $\deg_G(x)$ for the degree…

Let $\Gamma_n$ be the Fibonacci cube on binary strings of length $n$ with no consecutive $1$'s, let $F_n$ denote the $n$th Fibonacci number, and let $\operatorname{gp}_{e}(G)$ be t…

Let $G$ be a graph, and let ${\rm gp}_{\rightarrow}(G)$ denote the minimum general position number over all orientations of $G$. The lower-number-two hardness conjecture. It is NP-…

Let $G$ be a graph, and let ${\rm gp}_{\rightarrow}(G)$ be the minimum general position number over all orientations of $G$. The lower-number hardness conjecture. Determining the l…

Let $G$ be a graph, and let … be its general position spectrum. The interval-spectrum conjecture. The set $S_{{\rm gp}}(G)$ is an interval of integers. The spectrum is an interval…

Let $G$ be a connected graph of order $n\geq 3$. Define ${\rm gp}^{\rightarrow}(G)$ as the maximum general position number over all orientations of $G$, and…

Let $G$ be a graph, and let … be its general position spectrum. The nonconstant-spectrum conjecture. Every graph has two orientations with different general position numbers; equiv…

The exact lower-bound conjecture. The lower bound from Theorem is exact. This would in particular determine the value suggested by the computations for $k=3$, while the general cas…

Let $n$ and $d$ be positive integers, and let $\operatorname{Circ}(n,\{1,2,\ldots,d\})$ be the directed circulant graph whose arcs correspond to the generators $1,2,\ldots,d$. Let…

Terminal set existence conjecture. Every graph has a terminal set.

The lower general position product bound.

Let $\Gamma_n$ denote the Fibonacci cube of dimension $n$, let $F_n$ be the $n$th Fibonacci number, and let $\operatorname{gp}_{e}(\Gamma_n)$ denote its edge general position numbe…