The last fraction of a fractional conjecture

Abstract

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

A typo has been corrected in the introduction (concerning the citation of the result by Ito, Kennedy and Reed)