11 problems

Star-homology vanishing conjecture. Whenever $i\ge 2$ or $\lambda_2\ge 3$, the multiplicity of $\mathcal{S}_\lambda$ in $H_{i,0}(G)$ is $0$.

Star Specht-multiplicity conjecture. The multiplicity of $\mathcal{S}_{2^2 1^{n-4}}$ in $H_{1,0}(G)$ is

Planarity-detection conjecture. The chromatic symmetric homology $H_*(G;\mathbb{Z})$ can detect planarity according to whether $H_{1,0}(G;\mathbb{Z})$ has $\mathbb{Z}_2$-torsion, m…

Bidegree-(1,0) torsion conjecture. The graph $G$ is non-planar if and only if its chromatic symmetric homology in bidegree $(1,0)$ contains $\mathbb{Z}_2$-torsion.

Span bounds conjecture. The homological span satisfies

Chromatic symmetric homology interval conjecture. The groups $H_{i,0}(G;\mathbb{C})$ are non-trivial for every integer $i$ satisfying

Let $G$ be a graph. For each $i$, write $n_i$ for the rank of the free part of $H_i(G;\mathbb{Z})$, and suppose the nonzero homological degrees under consideration are $0,\ldots,k$…

Let $G$ be a graph with $n$ vertices and $m$ edges, and let $b$ be the number of blocks of $G$. Let $\operatorname{span}_0(G)$ denote the homological span of the degree-zero chroma…

Let $G$ be a graph, and let $\operatorname{span}_0(G)$ denote the homological span of its degree-zero chromatic symmetric homology. Nonvanishing conjecture. For every graph $G$, th…

Let $C_n$ be the cycle graph with $n$ vertices. Cycle homology conjecture. For $0\leq i\leq n-1$, $H_i(C_n;\mathbb{Z})$ is a free $\mathbb{Z}$-module of rank … The source supplies…

Let $G$ be a graph. Its chromatic symmetric homology has a $q$-degree, and $H_1(G;\mathbb{Z})$ denotes the homology in homological degree $1$ and $q$-degree zero. Planarity conject…