Plane graphs without cycles of length 4,6,74,6,7 or 8 are 3-colorable

In 1976, Steinberg conjectured that plane graphs without cycles of length 4 and 5 are 3-colorable. Recently, Borodin et al. proved that plane graphs without cycles of length from 4 to 7 are 3-colorable. In this note, we make a moderate improvement on the result of Borodin et al., that is, we prove that if a plane graph without cycles of length 4, 6 and 7 contains no adjacent 5-faces, then it is 3-colorable. As a corollary, every plane graph without cycles of length 4,6,74,6,7 and 8 is 3-colorable.