Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable
Combinatorics
2017-02-27 v1
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.
Cite
@article{arxiv.1702.07558,
title = {Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable},
author = {Ligang Jin and Yingli Kang and Michael Schubert and Yingqian Wang},
journal= {arXiv preprint arXiv:1702.07558},
year = {2017}
}
Comments
15 pages, 2 figure. arXiv admin note: text overlap with arXiv:1506.04629