9 problems

For every integer $n\ge 1$ and every $n$-vertex graph $G$ with independence number $\alpha(G)=2$, the graph $G$ contains $K_{\lceil n/2\rceil}$ as an odd minor.

Let $B(G)$ denote the bicircular matroid of a graph $G$, let $U_{t,t+2}$ be the uniform matroid of rank $t$ on $t+2$ elements, and let $M(K_t)$ and $M(K_t)^*$ be respectively the g…

Let $U_{2,q}$ be the $q$-element rank-$2$ uniform matroid, and more generally let $U_{t,q}$ be the $q$-element rank-$t$ uniform matroid. Let $M(K_t)$ be the graphic matroid of the…

Let $d \geq 2$. A graph is a forbidden minor for $d$-flattenability when it is a minimal obstruction to $d$-flattenability. For an edge $e$ of a graph $G$, $(G \setminus e,e)$ deno…

Let $G$ be a graph, let $f$ be a nonedge of $G$, and let $G \cup f$ denote the graph obtained by adding $f$. An atom of $G \cup f$ is a graph-theoretic atom containing $f$, and an…

DeVos–Kwon–Oum's branch-depth obstruction conjecture. For every positive integer $n$, there is an integer $d$ such that every matroid of branch-depth at least $d$ contains a minor…

Let $(G,\Sigma)$ be a signed bipartite graph, with unbalanced girth at least $2k$, that has no $(K_5,E(K_5))$-minor. The signed projective cube $SPC(2k-1)$ is the signed graph on v…

Bounded geodecity conjecture. For every graph $G$ there exists an integer $m$ such that every subdivision of $G$ lies in \H_m.

Let $m$ be a positive integer, and let $M$ be a $3$-connected clonal matroid. The matroids $U_{2,2m}$ and $U_{2m-2,2m}$ are uniform matroids, while $\Lambda_m$ and $\Delta_m$ denot…