10 problems

Let $G$ be a graph with maximal degree $d$, and let $K_n$ be the complete graph on $n$ vertices. Babson–Kozlov's conjecture. The complex $\operatorname{Hom}(G,K_n)$ is at least…

Hom-complex realization conjecture. If

Diameter-one conjecture. If

Let $G$ be the complement of the $1$-skeleton of a flag simplicial PL sphere, and let $s$ be the maximal valency of $G$. The associated graph coloring manifolds are the Hom complex…

Let $C_{2r+1}$ be an odd cycle, let $K_n$ be the complete graph on $n$ vertices, and let $\varpi_1^{n-2}(\text{\tt Hom}(C_{2r+1},K_n))$ denote the indicated Stiefel-Whitney charact…

A graph is a homotopy test graph if its Hom-complexes detect graph colorability through the corresponding homotopy-connectivity bound. Homotopy-test-graph conjecture. Every connect…

Björner–Lovász conjecture. If $\operatorname{Hom}(G,H)$ is $k$-connected, then

Let $G$ be a graph, let $d$ be its maximal valency, and let $K_n$ be the unlooped complete graph on $n$ vertices. For an integer $k\geq -1$, write $\text{Hom}(G,K_n)$ for the Hom c…

Bj{"o}rner–Lovász conjecture. The quantity

Let $X$ be a finite simplicial complex, let $T$ be a connected graph of diameter $1$, and let $G_{1,X}$ denote the graph associated with $X$ in the source. Dochtermann's conjecture…