Every planar graph without 5-cycles and [formula omitted] and adjacent 4-cycles is [formula omitted]-colorable

Abstract

In 1976, Steinberg conjectured that every planar graph without 4-cycles and 5-cycles is 3-colorable, and in 2003, Borodin and Raspaud further conjectured that every planar graph without 5-cycles and K4− is 3-colorable. Both conjectures are disproved in 2016 by Cohen-Addad et al. In this paper, we prove a relaxation of the conjectures that every planar graph without 5-cycles and K4− and adjacent 4-cycles is (2,0,0)-colorable, which improves the results of Chen et al. (2016) and of Liu et al. (2015).