The last fraction of a fractional conjecture

International audienceReed conjectured that for every ε>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+ε. 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