Local Shearer bound
Anders Martinsson, Raphael Steiner
TL;DR
This work provides a local strengthening of Shearer’s bound for triangle-free graphs by constructing a probability distribution on independent sets where each vertex v appears with probability (1−o(1))·ln d_G(v)/d_G(v). The authors develop a vertex-weighted generalization (Theorem aux) using a function f with specific convexity and differential properties, proven by induction and a perturbation argument, which then yields several sharp corollaries. They derive (i) a local bound implying improved fractional coloring bounds, (ii) the asymptotically tight bound χ_f(G) ≤ (√2+o(1))√(n/ln n) for n-vertex triangle-free graphs, and (iii) analogous bounds in terms of edges with a precise constant, as well as (iv) a spectral bound χ_f(G) ≤ (1+o(1))·ρ(G)/ln ρ(G). These results resolve conjectures of Kelly–Postle and Cames van Batenburg et al., and advance connections between local independence, fractional coloring, and spectral graph theory in triangle-free graphs.
Abstract
We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every vertex $v$ of $G$ is contained in a random independent set drawn from the distribution with probability $(1-o(1))\frac{\ln d(v)}{d(v)}$. This resolves the main conjecture raised by Kelly and Postle (2018) about fractional coloring with local demands, which in turn confirms a conjecture by Cames van Batenburg et al. (2018) stating that every $n$-vertex triangle-free graph has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{\frac{n}{\ln(n)}}$. Addressing another conjecture posed by Cames van Batenburg et al., we also establish an analogous upper bound in terms of the number of edges. To prove these results we establish a more general technical theorem that works in a weighted setting. As a further application of this more general result, we obtain a new spectral upper bound on the fractional chromatic number of triangle-free graphs: We show that every triangle-free graph $G$ satisfies $χ_f(G)\le (1+o(1))\frac{ρ(G)}{\ln ρ(G)}$ where $ρ(G)$ denotes the spectral radius. This improves the bound implied by Wilf's classic spectral estimate for the chromatic number by a $\ln ρ(G)$ factor and makes progress towards a conjecture of Harris on fractional coloring of degenerate graphs.