Steinberg's conjecture (1976) asserts that every planar graph without 4- and 5-cycles is 3-colourable. Despite several attempts, the conjecture remains unsettled, and the relaxations of the problem including avoiding a certain collection of cycles of short lengths are considered. Among many other things, it is known that planar graphs without adjacent triangles and without 5- and 7-cycles are 3-colourable. We make an improvement to this result by showing that planar graphs without 5-cycles and without triangles adjacent to 3- and 6-cycles are 3-colourable.
Joint work with Babak Farzad.