an:06804421
Zbl 1378.05073
Song, Hongjie; Xu, Changqing
Neighbor sum distinguishing total coloring of planar graphs without 4-cycles
EN
J. Comb. Optim. 34, No. 4, 1147-1158 (2017).
00371378
2017
j
05C15 05C10
neighbor sum distinguishing total coloring; combinatorial nullstellensatz; planar graph
Summary: Let \(G=(V,E)\) be a graph and \(\phi:V\cup E\rightarrow \{1,2,\dots,k\}\) be a proper total coloring of \(G\). Let \(f(v)\) denote the sum of the color on a vertex \(v\) and the colors on all the edges incident with \(v\). The coloring \(\phi\) is neighbor sum distinguishing if \(f(u)\neq f(v)\) for each edge \(uv\in E(G)\). The smallest integer \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number of \(G\), denoted by \(\chi^{\prime\prime}_\Sigma(G)\). \textit{M. Pil??niak} and \textit{M. Wo??niak} [Graphs Comb. 31, No. 3, 771--782 (2015; Zbl 1312.05054)] conjectured that \(\chi^{\prime\prime}_\Sigma(G)\leq \Delta (G)+3\) for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that \(\chi^{\prime\prime}_\Sigma(G)\leq\max\{\Delta (G)+2,10\}\) for planar graph \(G\) without 4-cycles. The bound \(\Delta (G)+2\) is sharp if \(\Delta (G)\geq 8\).
Zbl 1312.05054