Abstract
A total-k-coloring of a graph G is a mapping \(c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}\) such that any two adjacent or incident elements in \(V(G)\cup E(G)\) receive different colors. For a total-k-coloring of G, let \(\sum _c(v)\) denote the total sum of colors of the edges incident with v and the color of v. If for each edge \(uv\in E(G)\), \(\sum _c(u)\ne \sum _c(v)\), then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by \(\chi _{\Sigma }^{''}(G)\). Pilśniak and Woźniak conjectured \(\chi _{\Sigma }^{''}(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that for any planar graph G with maximum degree \(\Delta (G)\), \(ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}\), where \(ch^{''}_{\Sigma }(G)\) is the neighbor sum distinguishing total choosability of G.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (11101243, 11371355, 11471193), the Fundamental Research Funds of Shandong University and Independent Innovation Foundation of Shandong University (IFYT14012).
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Qu, C., Wang, G., Yan, G. et al. Neighbor sum distinguishing total choosability of planar graphs. J Comb Optim 32, 906–916 (2016). https://doi.org/10.1007/s10878-015-9911-9
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DOI: https://doi.org/10.1007/s10878-015-9911-9
Keywords
- Neighbour sum distinguishing total choosability
- Planar graph
- Total coloring
- Discharging
- Combinatorial Nullstellensatz