Planar graphs without 3-cycles at distance less than 4 and without 5-cycles are proved to be 3-colorable. We conjecture that, moreover, each plane graph with neither 5-cycles nor intersecting 3-cycles is 3-colorable. In this conjecture, none of the two assumptions can be dropped because there exist planar 4-chromatic graphs without 5-cycles, as well as planar 4-chromatic graphs without intersecting triangles. r
The first one says that any planar graph without triangles at distance less than 4 and without 5-cycles is 3-colorable (see REF ).
A sufficient condition for planar graphs to be 3-colorable
J. Comb. Theory, Ser. B
J. Comb. Theory, Ser. B
Mathematics Computer Science