11/4-colorability of subcubic triangle-free graphs

Fractional coloring of graphs is a natural linear relaxation of the classical notion of proper coloring. It has a number of equivalent formulations, either discrete, polyhedral, or probabilistic.
The fractional chromatic number of a graph $G$ is the supremum $x$ such that there is a probability distribution over the independent sets of $G$, such that each vertex has probability at least $x$ to be in a random independent set from this distribution. It can be equivalently defined as the supremum $x$ such that the vector with all entries equal to $1/x$ lies in the independent set polytope. Finally, this is also the supremum ratio $p/q$, (for some integers $p$ and $q$) such that $G$ has a coloring with $p$ colors in which every vertex is assigned a set of $q$ colors, and adjacent vertices receive disjoint sest of colors.

Every graph with chromatic number at most $k$ has fractional chromatic number at most $k$, and every $n$-vertex graph with fractional chromatic number at most $k$ has an independent set of size at least $n/k$. So in some sense, having bounded fractional chromatic number is an intermediate property between having bounded chromatic number and having a linear size independent sets.

The paper is related to a conjecture of Heckman and Thomas (2001), stating that triangle-free graphs of maximum degree 3 have fractional chromatic number at most 14/5, which is tight (due to two examples on 14 vertices). The conjecture was proved by Dvořák, Sereni, and Volec in 2011. It was then conjectured by Cames van Batenburg, Goedgebeur, and Joret that if we exclude these two examples, and four others, the fractional chromatic number can be decreased to 8/3 (which is tight for infinitely many graphs).

The authors prove that apart from the two examples on 14 vertices, all connected triangle-free graphs of maximum degree 3 have fractional chromatic number at most 11/4, which is best possible (if no further examples are excluded) and can be seen as the first step towards the conjecture of Cames van Batenburg, Goedgebeur, and Joret.

As the two excluded examples are non planar, their result implies that all triangle-free planar graphs of maximum degree 3 have fractional chromatic number at most 11/4. This improves upon the best known bound and is a significant step towards another conjecture of Heckman and Thomas (2006), stating that these graphs have fractional chromatic number at most 8/3.