7 problems

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

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

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…

Quadratic embeddability conjecture. Assume $1\leq\alpha\leq\beta\leq\gamma$ and $\beta\geq2$. Then $\Theta(\alpha,\beta,\gamma)$ is of QE class if and only if either

Let $\Theta_{p,p+2,p}$ be the theta graph formed by three internally disjoint paths of lengths $p$, $p+2$, and $p$, respectively, where $p\geq 2$. Its metric dimension is denoted b…

For positive integers $\ell$ and $t$, a $t$-theta graph is a graph consisting of two vertices joined by $t$ internally vertex-disjoint paths. A long balanced $t$-theta is an induce…

Let $k\geq3$, let $m$ be sufficiently large, and let $G$ be a graph of size $m$ with no isolated vertices. Let $C_t^{\triangle}$ denote the graph obtained from $C_t$ and $C_3$ by i…