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.
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.
We conjecture that our lower bound can be improved to $2$ and that the analogous upper bound holds as well.
There exists a constant $C > 0$ such that the following holds for all sufficiently large $d$ and all $C \le f \le d2 + 1$: \begin{enumerate} \item Every $d$-degenerate graph $G = (V, E)$ that admits a left $\frac{d2}{f}$-locally-sparse degeneracy ordering satisfies $\chi_f(G) \le (2 + o(1))\frac{d}{\log f}$. \item There exists a $d$-degenerate graph $G = (V, E)$ that admits a left $\frac{d2}{f}$-locally-sparse degeneracy ordering satisfying $\chi_f(G) \ge (2 - o(1))\frac{d}{\log f}$. \end{enumerate}