Neighbor sum distinguishing total colorings of planar graphs
A total [k]-coloring of a graph \(G\) is a mapping \(\phi : V (G) \cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}\) such that any two adjacent or incident elements in \(V (G) \cup E(G)\) receive different colors. Let \(f(v)\) denote the sum of the color of a vertex \(v\) and the colors of all incident edges of \(v\). A total \([k]\)-neighbor sum distinguishing-coloring of \(G\) is a total \([k]\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(f(u)\ne f(v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak conjectured \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that this conjecture holds for any planar graph with maximum degree at least 13.
neighbor sum distinguishing total coloring planar graph maximum degree