16 problems

Let $G$ be a bipartite circle graph, namely a circle graph that is bipartite, and let $I(G)$ be its independence complex, whose simplices are the subsets of pairwise non-adjacent v…

Let $G$ be a circle graph, namely the intersection graph of the chords in a chord diagram, and let $I(G)$ be its independence complex, whose simplices are the subsets of pairwise n…

Przytycki–Silvero conjecture for bipartite circle graphs. If $G$ is a bipartite circle graph, then $I(G)$ has the homotopy type of a wedge of spheres.

Przytycki–Silvero conjecture. If $G$ is a circle graph, then $I(G)$ has the homotopy type of a wedge of spheres.

Exact homology-rank conjecture. The rank of the $p$-dimensional homology group of $\operatorname{Ind}(\operatorname{KG}(n,k))$ is

Let $I(G)$ denote the independence complex of a graph $G$, let $K_n$, $K_m$, and $K_l$ be complete graphs, and let $f(n,m,l)$ denote the unspecified number of spheres occurring in…

Polytope characterization conjecture. There is a polytope $P$ whose vertices are labelled by the edges of $G$ and whose faces are in bijection with the induced subgraphs of $G$ hav…

Engström's sphere-or-contractible conjecture. For any graph $G$, we have that $\mathrm{Ind}(H)$ is contractible or homotopy equivalent to a sphere for every induced subgraph $H$ of…

Kalai–Meshulam total Betti number conjecture. For any graph $G$, we have

Kalai–Meshulam Euler characteristic conjecture. For any graph $G$, we have

For $m,n\geq 2$, let $G_{m,n}$ be the rectangular grid graph with vertex set … and edges joining $(i,j)$ to $(i+1,j)$ or $(i,j+1)$ whenever those vertices exist. For an integer $r$…

Let $K_n$ denote the complete graph on $n$ vertices, let $\times$ denote the graph product, and let ${\rm{Ind}}(G)$ denote the independence complex of a graph $G$. For $n\geq 2$, t…

Let $P_m$ be the path graph on $m$ vertices and $C_n$ the cycle graph on $n$ vertices. Define … where $Z(G)=1-\chi(\operatorname{Ind}(G))$ is the Witten index associated with the i…

Generating-function denominator conjecture. For every $n\geq 0$, there are polynomials $h_n(t)$ such that

Let $C_{m,n}$ be the square-grid cylinder, let $Q_2$ be the class of configurations specified in the surrounding construction, and let $\pi$ be the associated projection map. For…

Let $C_{m,n}$ be the square-grid cylinder, let $Z(C_{m,n})$ be the hard-particle partition function at activity $-1$, and suppose that $n$ is odd. Jonsson's cylinder partition-func…