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Listed as No. 53 among the one hundred famous unsolved problems in [J. A.Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008] is Steinberg'sconjecture, which states that every planar graph without 4- and 5-cycles is3-colorable. In this paper, we show that plane graphs without 4- and 5-cyclesare 3-colorable if they have no ext-triangular 7-cycles. This implies that (1)planar graphs without 4-, 5-, 7-cycles are 3-colorable, and (2) planar graphswithout 4-, 5-, 8-cycles are 3-colorable, which cover a number of known resultsin the literature motivated by Steinberg's conjecture.

Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable