Weak-Dynamic Coloring of Graphs Beyond-Planarity

A weak-dynamic coloring of a graph is a vertex coloring (not necessarily proper) in such a way that each vertex of degree at least two sees at least two colors in its neighborhood. It is proved that the weak-dynamic chromatic number of the class of k-planar graphs (resp. IC-planar graphs) is equal to (resp. at most) the chromatic number of the class of 2k-planar graphs (resp. 1-planar graphs), and therefore every IC-planar graph has a weak-dynamic 6-coloring (being sharp) and every 1-planar graph has a weak-dynamic 9-coloring. Moreover, we conclude that the well-known Four Color Theorem is equivalent to the proposition that every planar graph has a weak-dynamic 4-coloring, or even that every \(C_4\)-free bipartite planar graph has a weak-dynamic 4-coloring. It is also showed that deciding if a given graph has a weak-dynamic k-coloring is NP-complete for every integer \(k\ge 3\).