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arXiv · 1702.07558

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

Ligang Jin·Yingli Kang·Michael Schubert·Yingqian Wang
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Abstract

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's conjecture, which states that every planar graph without 4- and 5-cycles is 3-colorable. In this paper, we show that plane graphs without 4- and 5-cycles are 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 graphs without 4-, 5-, 8-cycles are 3-colorable, which cover a number of known results in the literature motivated by Steinberg's conjecture.

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Ligang Jin, Yingli Kang, Michael Schubert, Yingqian Wang. 2017-02-24. Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable. https://arxiv.org/abs/1702.07558

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