The last fraction of a fractional conjecture

TL;DR

This paper proves Reed's fractional total chromatic conjecture for all cases, confirming that graphs with large girth and bounded degree have fractional total chromatic number close to their maximum degree plus one.

Contribution

It completes the proof of Reed's conjecture by establishing the result for all degrees, including the remaining cases not previously proven.

Findings

Reed's conjecture is true for all degrees and girth conditions.

Graphs with large girth and bounded degree have fractional total chromatic number at most Δ+1+ε.

The paper resolves the conjecture for the remaining cases.

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.