Planar graphs without cycles of length 4 or 5 are (11:3)-colorable

TL;DR

This paper proves that planar graphs lacking 4- and 5-length cycles are (11:3)-colorable, establishing a new coloring property and confirming a weakened form of Steinberg's conjecture.

Contribution

It demonstrates that such planar graphs are (11:3)-colorable, a significant relaxation of Steinberg's conjecture, and provides bounds on their independent set sizes.

Findings

Planar graphs without 4- or 5-cycles are (11:3)-colorable.

Such graphs have an independent set of size at least 3n/11.

The result weakens Steinberg's conjecture.

Abstract

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.