26 problems

Let $T$ be a tree, let $\Delta(T)$ denote its maximum degree, and say that $T$ is $e$-positive when its chromatic symmetric function $X_T$ has a nonnegative expansion in the elemen…

Wang's conjecture. All theta graphs $\theta_{abc}$ are $e$-positive.

Twin-free reduction conjecture. If $P$ is $(3+1)$-free and twin-free, then $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive.

Foley–Hoàng–Merkel's twinning conjecture. If $P$ is $(3+1)$-free and $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive, then $X_{\operatorname{Inc}(P_v)}(\mathbf{x})$ is $e$-…

For positive integers $m,n$, let $S(a,b,c)$ denote the spider formed from three paths of lengths $a,b,c$ with a common end. A graph is $e$-positive when its chromatic symmetric fun…

For a partition $ambda=ambda_1\dotsmambda_d$ of $n-1$ with $d\ge 3$, the spider $S(\lambda)$ is the $n$-vertex tree consisting of paths of lengths $\lambda_1,\dots,\lambda_d$ with…

For any integers $a\ge b\ge c\ge 1$, let $\theta_{abc}$ be the theta graph formed from two distinct vertices joined by three pairwise disjoint paths of lengths $a$, $b$, and $c$. A…

Let $\mathbf{m}\colon[n]\to[n]$ be a Hessenberg function, let $G_\mathbf{m}$ be its indifference graph, and let $g_k(G_\mathbf{m};x,q)$ be the symmetric functions defined in the pa…

Let $(G,\omega)$ be a set-weighted graph with $n$ vertices and total weight $d$, and write … Let $\mu\leq d$ be an integer, viewed as a one-part partition, that is $s$-allowable in…

Rooted $(3+1)$-free $e$-positivity conjecture. For every $r\in P$, if $P-r$ is $(\mathbf{3}+\mathbf{1})$-free, then

Let $\gamma$ and $\lambda$ be the parameters used to define the set $\operatorname{PF}^{\gamma}_{\lambda}$ of $\gamma$-parking functions, let $S$ be a subset of the area cells of…

Let $F$ be a symmetric function with a positive monomial-basis expansion, let $G$ be a symmetric function with a positive Schur-basis expansion, and let $\Delta_F$ and $\Xi$ denote…

Spider e-positivity conjecture. If $b$ is even and $b\not\equiv2\pmod3$, then the spider $S(b^2+b,\,b,\,1)$ is $e$-positive.

Dahlberg–Shepherd conjecture. Any tree with a vertex of degree at least $4$ is not $e$-positive.

Appendable $(e)$-positivity conjecture. Every labelled unit interval graph is appendable $(e)$-positive.

Dahlberg's conjecture. All labelled unit interval graphs are $(e)$-positive.

Line-graph conjecture. If a spider $S$ is $e$-positive, then its line graph $S_L$ is $e$-positive.

Leg-combination conjecture. If $S$ is $e$-positive, then $S'$ is $e$-positive.

Aliniaeifard–van Willigenburg–Wang conjecture. The family of spiders

Let $P(\mathbf{m})$ be the natural unit interval order associated with a sequence $\mathbf{m}=(m_1,m_2,m_3,n,\ldots,n)$, and let $G$ be its incomparability graph. Let…

Let $P(\mathbf{m})$ be the natural unit interval order associated with a sequence $\mathbf{m}=(2,m_2,m_3,n,\ldots,n)$, and let $G$ be its incomparability graph. Let…

Induced claw-or-net conjecture. If $G$ is not $e$-positive, then $G$ contains an induced claw or an induced net.

Strongly e-positive graph characterization conjecture. A graph $G$ is strongly $e$-positive if and only if it is $(\text{claw},\text{net})$-free.

Let $\mathrm{DH}_{n,r}$ denote the multivariate diagonal harmonics space, and let $\Phi_{n,r}(q)$ be the Frobenius characteristic of its character, with parameter $q_r$. A symmetri…

Let $\Gamma$ be a circular unit arc digraph, and let $H$ be a graph obtained from $\Gamma$ by marking $k$ corner edges strict. Strict-corner difference conjecture. Then … is…