A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is (11:3)-colorable, a weakening of recently disproved Steinberg's conjecture. In particular, each such graph with n vertices has an independent set of size at least 3n/11.
@article{arxiv.1802.04179,
title = {Planar graphs without cycles of length 4 or 5 are (11:3)-colorable},
author = {Zdeněk Dvořák and Xiaolan Hu},
journal= {arXiv preprint arXiv:1802.04179},
year = {2019}
}