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Erschienen in:
01.10.2015
Neighbor sum distinguishing total colorings of planar graphs
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 3/2015
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Abstract
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 \(G\). Pilśniak and Woźniak 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.