3 problems

Nešetřil's universal target conjecture. For every integer $k$, there is a graph $H$ of girth at least $k$ and an integer $N$ such that every cubic graph $G$ of girth at least $N$ h…

Let $G$ be a graph of girth $g$, meaning that $g$ is the length of its shortest cycle, and suppose that every vertex of $G$ has degree at least $3$. Let $c_2(G)$ denote the least d…

Let $\Delta\ge5$ be an odd integer. Let the fractional total chromatic number be the fractional analogue of the total chromatic number, and let the girth of a graph be the length o…