Fractional total colourings of graphs of high girth

TL;DR

This paper proves Reed's conjecture for graphs with maximum degree 3 and even Delta>=4, showing that high girth graphs have fractional total chromatic number exactly Delta+1, confirming a long-standing hypothesis.

Contribution

The paper establishes the conjecture for specific degrees, demonstrating that high girth graphs achieve the fractional total chromatic number exactly equal to Delta+1.

Findings

Confirmed Reed's conjecture for Delta=3 and even Delta>=4.

High girth graphs have fractional total chromatic number exactly Delta+1.

Provides a stronger form of the conjecture for these cases.

Abstract

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1.