A sufficient condition for planar graphs to be 3-colorable
The authors prove that every proper 3-coloring of a face of size 3 or 7 in a connected planar graph without pairs of 3-cycles at distance less than 4 and without 5-cycles can be extended to a proper 3-coloring of the whole graph. Form this and Grötzsch's theorem, it follows that planar graphs without 3-cycles at distance less than 4 and without 5-cycles are 3-colorable. The authors conjecture that planar graphs without 5-cycles in which every two 3-cycles are vertex-disjoint (or every two 3-cycles are edge-disjoint) are 3-colorable.
scientific article; zbMATH DE number 1970673