Fractional total colorings of graphs of high girth

František Kardoš

Abstract

(joint work with Daniel Kral, Jean-Sebastien Sereni)

Reed conjectured that for every $\varepsilon > 0$ and every integer $\Delta$ there exists $g$ such that the fractional total chromatic number of every graph $G$ of maximum degree $\Delta$ and girth at least $g$ is at most $\Delta+1+\varepsilon$. The conjecture was proven to be true in a stronger form when $\Delta$ is three or an even integer by Kral, Kaiser and King. We settle the conjecture by proving that it holds also for the remaining cases, i.e. when $\Delta$ is an odd integer.