The last fraction of a fractional conjecture

Abstract

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most $\Delta+1+\varepsilon$. The conjecture was proven to be true when $\Delta=3$ or $\Delta$ is even. We settle the conjecture by proving it for the remaining cases.