8 problems

Hom-complex realization 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…

For a graph $G$, let $\text{\tt Hom}(G,H)$ denote its Hom-complex. Let $V_2(\mathbb R^{n-1})$ be the Stiefel manifold of orthonormal $2$-frames in $\mathbb R^{n-1}$. Csorba's conje…

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…

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…

For a bipartite graph of the form $K_2\times K_n$, let $d$ denote its maximal degree and let $K_m$ be a complete graph. The lower bound for the connectivity of…

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…