7 problems

Bounded-cycle-space fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}^2\to\mathbb{N}^2$ such that, for every graph $G$ whose cycle space is generated…

Let $G$ be an $n$-vertex graph, where $n$ is odd. Write $\kappa(G)$ for its vertex-connectivity, $\alpha(G)$ for its independence number, $\mathcal{C}(G)$ for its cycle space, and…

Generalized $k$-independence conjecture. For every $n$-vertex simple graph $G$,

A graph $G$ is toroidal if it can be embedded on the torus, and its basis number $\operatorname{bn}(G)$ is the smallest integer $k$ for which $G$ has a cycle-space basis in which e…

Let $X$ be an integral projective variety, and let $N_k(X)_{\mathbb{Z}}$ denote the group of numerical classes of $k$-cycles. A class is pseudo-effective if it lies in the pseudo-e…

The complete-partite graph cycle-space conjecture. For every $s\geq 4$,

Bondy's conjecture. If