A note on the Three Color Problem on planar graphs without 4-and 5-cycles and without ext-triangular 7-cycles

Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. This conjecture is disproved by constructing a planar graph with no cycles of length four or five but intersecting triangles. Jin et al. proved that plane graphs without 4-and 5-cycles and without ext-triangular 7-cycles are 3-colorable [SIAM J. Discrete Math. 31 (3) (2017) 1836-1847]. In this paper, we point out a mistake of their proof and give an improved proof.(c) 2022 Elsevier B.V. All rights reserved.