Random independent sets in triangle-free graphs

Abstract We establish several new results on the existence of probability distributions on the independent sets in triangle-free graphs where each vertex is present with a given probability. This setting was introduced and studied under the name of “fractional coloring with local demands” by Kelly and Postle and is closely related to the well-studied fractional chromatic number of graphs. Our first main result strengthens Shearer’s classic bound on independence number, proving that for every triangle-free graph G there exists a distribution over independent sets where each vertex v appears with probability $(1-o(1))\frac {\ln d_G(v)}{d_G(v)}$ , resolving a conjecture by Kelly and Postle. This in turn implies new upper bounds on the fractional chromatic number of triangle-free graphs with a prescribed number of vertices or edges, which resolves a conjecture by Cames van Batenburg et al. and addresses yet another one by the same authors. Our second main result resolves Harris’ conjecture on triangle-free d -degenerate graphs, showing that such graphs have fractional chromatic number at most $(4+o(1))\frac {d}{\ln d}$ . As previously observed by various authors, this in turn has several interesting consequences. A notable example is that every triangle-free graph with minimum degree d contains a bipartite induced subgraph of minimum degree $\Omega (\log d)$ . This settles a conjecture by Esperet, Kang, and Thomassé. The main technique employed to obtain our results is the analysis of carefully designed random processes on vertex-weighted triangle-free graphs that preserve weights in expectation. The analysis of these processes yields weighted generalizations of the aforementioned results that may be of independent interest.