Distinguishing colorings of 3-connected planar graphs with five colors (Q2799604)

A proper coloring of \(G\) with \(k\) colors is called a distinguishing \(k\)-coloring of \(G\) if there is no color-preserving automorphism of \(G\) other than the identity map. It is proved that every \(3\)-connected planar graph, with the exceptions of \(K_{2,2,2}\) and \(C_6+\overline K_2\), admits a distinguishing \(5\)-coloring which uses color \(5\) for one vertex.NEWLINENEWLINEBy contrast, it is given examples of \(3\)-connected planar graphs that have distinguishing \(4\)-colorings but no distinguishing \(4\)-coloring with one color used only once.