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Degenerate and star colorings of graphs on surfaces. (English) Zbl 1230.05133
Summary: We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of G. Fertin, A. Raspaud, and B. Reed [J. Graph Theory 47, No. 3, 163–182 (2004; Zbl 1055.05051)]: If \(G\) is a graph of maximum degree \(\Delta \), then \(G\) admits a degenerate star coloring using \(O(\Delta ^{3/2})\) colors. We use this result to prove that every graph of genus \(g\) admits a degenerate star coloring with \(O(g^{3/5})\) colors. It is also shown that these results are sharp up to a logarithmic factor.

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
Citations:
Zbl 1055.05051
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References:
[1] Albertson, M.O.; Chappell, G.G.; Kierstead, H.A.; Kündgen, A.; Ramamurthi, R., Coloring with no 2-colored \(P_4\)’s, Electron. J. combin., 11, 13, (2004) · Zbl 1053.05045
[2] Alon, N.; McDiarmid, C.; Reed, B., Acyclic colouring of graphs, Random structures algorithms, 2, 277-288, (1991) · Zbl 0735.05036
[3] Alon, N.; Mohar, B.; Sanders, D.P., On acyclic colorings of graphs on surfaces, Israel J. math., 94, 273-283, (1996) · Zbl 0857.05033
[4] Alon, N.; Spencer, J.H., The probabilistic method, (2000), Wiley New York
[5] Borodin, O.V., On decomposition of graphs into degenerate subgraphs in Russian, Diskret. analiz., 28, 3-12, (1976) · Zbl 0425.05058
[6] Borodin, O.V., A proof of grünbaum’s conjecture on the acyclic 5-colorability of planar graphs, Sov. math. dokl., 17, 1499-1502, (1976) · Zbl 0363.05031
[7] Borodin, O.V., On acyclic colorings of planar graphs, Discrete math., 25, 211-236, (1979) · Zbl 0406.05031
[8] Fertin, G.; Raspaud, A.; Reed, B., Star coloring of graphs, J. graph theory., 47, 163-182, (2004) · Zbl 1055.05051
[9] Grünbaum, B., Acyclic colorings of planar graphs, Israel J. math., 14, 390-412, (1973) · Zbl 0265.05103
[10] Kierstead, H.A.; Mohar, B.; Špacapan, S.; Yang, D.; Zhu, X., The two-coloring number and degenerate colorings of planar graphs, SIAM J. discrete math., 23, 1548-1560, (2009) · Zbl 1207.05058
[11] Mohar, B.; Thomassen, C., Graphs on surfaces, (2001), Johns Hopkins University Press Baltimore · Zbl 0979.05002
[12] Molloy, M.; Reed, B., Graph colouring and the probabilistic method, (2002), Springer · Zbl 0987.05002
[13] Nešetřil, J.; Ossona de Mendez, P., (), 651-664
[14] Nešetřil, J.; Ossona de Mendez, P., Grad and classes with bounded expansion I. decompositions, Europ. J. combinatorics., 29, 760-776, (2008) · Zbl 1156.05056
[15] Nešetřil, J.; Ossona de Mendez, P., From sparse graphs to nowhere dense structures: decompositions, independence, dualities and limits, European congress of mathematics, Eur. math. soc., 135-165, (2010), Zürich · Zbl 1196.05080
[16] Rautenbach, D., A conjecture of borodin and a coloring of grünbaum, J. graph theory., 58, 139-147, (2008) · Zbl 1154.05035
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