Abstract
A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without l cycles is at most Δ(G) + 2 if (k, l) ∈ {(6, 3), (7, 4), (6, 5), (7, 6)}.
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Supported by National Natural Science Foundation of China (Grant Nos. 10971121, 10631070, 60673059)
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Sun, X.Y., Wu, J.L. Acyclic total colorings of planar graphs without l cycles. Acta. Math. Sin.-English Ser. 27, 1315–1322 (2011). https://doi.org/10.1007/s10114-011-8640-y
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DOI: https://doi.org/10.1007/s10114-011-8640-y