2 problems

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…