11 problems

Let $G$ be a graph, and let $P_G(q)$ denote its chromatic polynomial, whose roots are called chromatic roots. The Beraha numbers are the numbers … A planar triangulation is a plana…

Maxmaxflow chromatic-root conjecture. There exist universal constants $C(\Lambda)<\infty$ such that every chromatic root of every loopless graph of maxmaxflow $\Lambda$ lies in

Asymptotic corank-growth conjecture. As $k\to\infty$,

Fixed-corank theta extremal conjecture. If $G\neq K_4$, then

Maxmaxflow conjecture. There exist universal constants $C(k)<\infty$ such that every chromatic root $z$ of any graph $G$ with $\Lambda(G)=k$ lies in the disc

Sharp series-parallel bound conjecture. Every chromatic root $z$ of $G$ lies in the disc

Series-parallel chromatic-root bound. There exists a universal constant $C<\infty$ such that, for every series-parallel graph $G$ of maximum degree $k$, every chromatic root $z$ li…

Extremal generalized-theta conjecture. For every $k$, the $k$-ary generalized theta graph whose chromatic root $z$ maximizes $|z-1|$ is $\Theta^{(2,k)}$, the graph with all path le…

The n-alpha conjecture. The set $R$ of all chromatic roots is closed under multiplication by positive integers; equivalently, if $\alpha\in R$ and $n$ is a positive integer, then…

Let $S_{m,n}$ be the graph family considered in the paper, and let a chromatic root mean a zero of its chromatic polynomial in the complex $q$-plane. There exists a finite constant…

Let $G$ be a 3-connected graph that is not bipartite of odd order. A chromatic root is a real or complex zero of the chromatic polynomial $P(G,x)$. Jackson's conjecture. The graph…