@article{Liang2026, 
author = {Zuosong Liang and Danzhang Liao and Chunsong Bai},
title = {On the    3-coloring of planar graphs without cycles of length from    4 to    6},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {12895-12909},
keywords = {planar graphs, 3-coloring, Steinberg's conjecture},
url = {https://www.sciopen.com/article/10.3934/math.2026530},
doi = {10.3934/math.2026530},
abstract = {In    1976, Steinberg conjectured that every planar graph without    4- and    5-cycles is    3-colorable. This conjecture was proved false by Cohen-Addad et al in 2017. Erdős raised the following question: Is there an integer    k such that every planar graph without cycles of length from    4 to    k is    3-colorable? Borodin et al. proved that every planar graph without cycles of length from    4 to    7 is    3-colorable [Planar graphs without cycles of length from    4 to    7 are    3-colorable, J. Combin.Theory Ser. B, 93 (2005), 303–311]. However, the question whether every planar graph without cycles of length from    4 to    6 is    3-colorable is not answered yet and full of challenges. A    7-cycle is called a special    7-cycle if it shares an edge with another    7-cycle or    9-cycle. In this paper, we prove that every planar graph without cycles of length from    4 to    6 and without special    7-cycles is    3-colorable which is an improvement of Borodin's result.}
}