12 problems

Let $G=(V,E)$ be a unit interval graph, with its vertices in the natural order. Define its chromatic quasisymmetric function by … where the sum is over proper colorings and…

Let $G$ be a natural unit interval graph, and write its chromatic quasisymmetric function in the elementary basis as … Here $c_\mu(q)$ is the coefficient of $e_\mu(\mathbf{x})$ in…

First-preserving involution conjecture. Every natural unit interval graph, with a specific ordering of its edges, has a first-preserving involution.

Max-load extension conjecture. Under these assumptions,

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

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

Let $G$ be a graph on $n$ vertices, let $w_G$ be its associated graphon, and let $\widetilde{\Gamma}(G)$ and $\widetilde{\Gamma}(w_G)$ be the graph and graphon parameters defined b…

First-increase inequalities. One has

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…

Let $m=m_1\cdots m_n$ satisfy $1\leq m_1\leq\cdots\leq m_n\leq n$ and $m_i\geq i$ for each $i$, and let $G(m)$ be the graph on $[n]$ with an edge $(i,j)$ exactly when $i<j\leq m_i$…

Reflection conjecture. If $\Gamma$ is a connected unit interval graph limit, then $\overline\Psi^{-1}(\Gamma)$ consists of either one semiorder limit $\Pi$, with $\Pi^\dagger=\Pi$,…