Let G be a graph that can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that the chromatic number of G does not exceed 7. The bound \( \chi (G)\leq \frac{{9+\sqrt{17+64g }}}{2} \) is also proved for a graph G that can be drawn on a surface of genus g (g ≥ 1) in such a way that any edge intersects at most one other edge. Bibliography: 8 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 406, 2012, pp. 95–106.
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Nenashev, G.V. A Bound on the Chromatic Number of an Almost Planar Graph. J Math Sci 196, 784–790 (2014). https://doi.org/10.1007/s10958-014-1693-6
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DOI: https://doi.org/10.1007/s10958-014-1693-6