Geodesic
On graphs which can be drawn on an orientable surface with small number of intersections on an edge
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VII, Tome 427 (2014), pp. 114-124 
O. E. Samoilova. On graphs which can be drawn on an orientable surface with small number of intersections on an edge. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VII, Tome 427 (2014), pp. 114-124. http://geodesic.mathdoc.fr/item/ZNSL_2014_427_a7/
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     author = {O. E. Samoilova},
     title = {On graphs which can be drawn on an orientable surface with small number of intersections on an edge},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_427_a7/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $k$ and $g$ be nonnegative integers. We call a graph $k$-nearly $g$-spherical, if it can be drawn on an orientable surface of genus $g$ such that each edge intersects at most $k$ other edges in inner points. It is proved that for $k\leq4$ the number of edges of a $k$-nearly $g$-spherical graph on $v$ vertices does not exceed $(k+3)(v+2g-2)$. It is also proved that the chromatic number of a $k$-nearly $g$-spherical graph does not exceed $\frac{2k+7+\sqrt{4k^2+12k+1+16(k+3)g}}2$.

[1] J. Pach, G. Toth., “Graphs drawn with few crossing per edge”, Combinatorica, 17:3 (1997), 427–439 | DOI | MR | Zbl

[2] P. J. Heawood, “Map colour theorem”, Quart. J. Math., 24 (1890), 332–338 | Zbl

[3] D. V. Karpov, “Verkhnyaya otsenka kolichestva reber v dvudolnom pochti planaronom grafe”, Zap. nauchn. semin. POMI, 406, 2012, 12–30 | MR

[4] G. V. Nenashev, “Otsenka khromaticheskogo chisla pochti planarnogo grafa”, Zap. nauchn. semin. POMI, 406, 2012, 95–106 | MR