4 problems

Circular homomorphism conjecture. Every planar graph $G$ of girth at least $4k$ admits a homomorphism to $C_{2k+1}$. Equivalently,

Nešetřil's Pentagon Conjecture. If $G$ is a cubic graph of sufficiently high girth, then $G$ is homomorphic to $C_5$.

Let $G_{n,\frac{c}{n}}$ be the random graph with edge probability $\frac{c}{n}$, and let $\chi_{\mathrm{c}}(G)$ denote its circular chromatic number. Density conjecture. There are…

Let ${\rm KG}(m,n)$ denote the Kneser graph, whose vertices are the $n$-element subsets of an $m$-element set, with two vertices adjacent when the corresponding subsets are disjoin…