A new proof of the independence ratio of triangle-free cubic graphs

A Fractional Analogue of Brooks' Theorem

“…In the case ∆(G) = 3, Heckman and Thomas [4] conjectured that χ f (G) ≤ 14/5 if G is triangle-free. Hatami and Zhu [3] proved χ f (G) ≤ 3 − 3 64 for any triangle-free graph G with ∆(G) = 3.…”

“…Theorem 2 implies f (k) > 0 for all k ≥ 3. Reed's result [8] implies f (k) ≥ 1 for sufficiently large k. Heckman and Thomas [4] conjectured f (3) = 1/5. It is an interesting problem to determine the value f (k) for small k. Is f (4) = 1 3 ?…”

The fractional chromatic number of triangle-free subcubic graphs

“…Thus, we may write, for instance, that a template ∆ is covered by pairs (x, y) 2 and (z, z) 4 . By Lemma 7, we then have…”

“…The event C 1 AD + is covered by the pair (u − , u − ) 4 , so by Lemma 7 its probability is P(C 1 AD + ) ≥ 79/80·0.25/256 > 0.24/256. The event C 1 AD 0 has up to two sensitive pairs: it is covered by (u − , u − ) 4 and (u +2 , v − ) 2 , where the latter pair is 2-free because uZv is not short. We obtain P(…”

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

“…We point out that the motivation for the formulation of Theorem 3.2 as well as for some parts of its proof comes from the work of Heckman and Thomas [10], in which an analogous strengthening is used to prove that every subcubic triangle-free graph on n vertices contains an independent set of size at least 5n/14.…”