Abstract
A k-coloring of a map M(G) on a closed surface with underlying graph G is said to be distinguishing if no automorphism of M(G) other than the identity map preserves the colors given by the coloring. In particular, if there is a distinguishing k-coloring of M(G) which uses color k at most once, then M(G) is said to be nearly distinguishing (\(k{-}1\))-colorable. We shall show that any 3-regular map on a closed surface is nearly distinguishing 3-colorable unless it is one of the three exceptions.
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Negami, S. 3-Regular Maps on Closed Surfaces are Nearly Distinguishing 3-Colorable with Few Exceptions. Graphs and Combinatorics 31, 1929–1940 (2015). https://doi.org/10.1007/s00373-015-1620-8
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DOI: https://doi.org/10.1007/s00373-015-1620-8