4 problems

Seymour's conjecture. Every planar graph whose odd cycles all have length at least $2k+1$ has a homomorphism to $PQ_{2k}$.

Let $r$ and $k$ be integers with $r\geq k$. Consider the simplicial complex associated to the graph $\mathcal{PC}_{2r}^{(2k-1)}$, where $\mathcal{PC}_{2r}^{(2k-1)}$ is the $(2k-1)$…

Let $r$ and $k$ be integers with $r\geq k$. For a graph $G$ and a positive integer $l$, let $G^{(l)}$ be the graph on the same vertex set in which two vertices are adjacent when th…

Let $r$ and $k$ be integers with $r\geq k$, and let $\mathcal{PC}_{2r}$ and $\mathcal{PC}_{2k}$ denote projective cubes. A graph homomorphism is a map preserving adjacency. Surject…