27 problems

Goyal–Shukla–Singh's conjecture. The independence complex of $K_2 \times K_3 \times K_n$ is homotopy equivalent to

Let $G$ be a simple undirected graph, and let $I(G)$ be its independence complex, whose simplices are the independent vertex sets of $G$. Say that $G$ has no induced cycles of leng…

Let $G=(V,E)$ be a graph, and let $\operatorname{Ind}(G)$ denote its independence complex. Define $\psi$ recursively by … Here $N(e)$ denotes the set of vertices adjacent to an end…

Let $G$ be a cylindrical square grid graph, and let $I(G)$ denote its independence complex, whose simplices are the independent subsets of $V(G)$. Iriye's conjecture. The geometric…

Fendley's conjecture. All eigenvalues of $\mathbb{O}_K$ are roots of unity.

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…

Let $C_n$ and $C_k$ be cycle graphs with $n$ and $k$ vertices, respectively, and let $I(C_n \times C_k)$ denote the independence complex of their Cartesian product. Write…

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…

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 $G$ be a graph, and let $I(G)$ denote its independence complex. Say that $G$ has the property that, for every induced subgraph $H$, the Euler characteristic of $I(H)$ has modul…

Finiteness conjecture. For every finite wedge sum of spheres $X$, there are only finitely many chordal graphs $G$ such that $G$ has no good pair and

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 a square-grid cylinder and $I(C_{m,n})$ its independence complex. Assume that $n$ is odd and that $Z(C_{m,n})=1$. Simplicity conjecture. The independence complexes…

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…

Let $C_{m,n}$ be the square-grid cylinder and let $I(C_{m,n})$ denote its independence complex. For $i\geq0$, write $H_i(I(C_{m,n}))$ for its $i$-th homology group. Homology symmet…

Let $m$ and $n$ be relatively prime, and consider the hard-square model on a square grid with toroidal identifications. For its independent sets $\sigma$, form the alternating sum…