Optimal leading constant for fractional chromatic number of triangle-free degenerate graphs

Determine the optimal leading constant C in the asymptotic relationship between the fractional chromatic number and degeneracy of triangle-free graphs, specifically by proving that, for every sufficiently large d, (i) every d-degenerate triangle-free graph G satisfies χ_f(G) ≤ (1 + o(1))d/ln d, and (ii) some d-degenerate triangle-free graph G satisfies χ_f(G) ≥ (1 − o(1))d/ln d.

Background

The paper proves Harris’s conjecture that every d-degenerate triangle-free graph has fractional chromatic number O(d/ln d), obtaining the upper bound (4 + o(1))d/ln d. Combining this result with a known lower-bound construction gives that the optimal leading constant C lies between 2 and 4.

The authors identify C = 1 as the most plausible value and formulate Conjecture 1.4 in two parts: an asymptotically sharp universal upper bound for all d-degenerate triangle-free graphs and a matching lower-bound construction. Establishing both parts would determine the exact asymptotic leading constant.

References

Finally, given the resolution of Harris’ conjecture, a natural remaining question is to determine the optimal leading constant C for the problem. In particular, by combining Theorem 1.2 with [4], we know that 2 ≤ C ≤ 4. It would appear that the most reasonable answer is C = 1. We state this as a conjecture. Conjecture 1.4. The following holds for any sufficiently large d. (i) χf (G) ≤ (1 + o(1)) d/ln d for all d-degenerate triangle-free graphs G. (ii) There exists a d-degenerate triangle-free graph G such that χf (G) ≥ (1 − o(1)) d/ln d.

— Triangle-free $d$-degenerate graphs have small fractional chromatic number  (2501.18238 - Martinsson, 30 Jan 2025) in Conjecture 1.4, Section 1 (Introduction), p. 3