an:06517435
Zbl 1331.05084
Qu, Cunquan; Wang, Guanghui; Wu, Jianliang; Yu, Xiaowei
On the neighbor sum distinguishing total coloring of planar graphs
EN
Theor. Comput. Sci. 609, Part 1, 162-170 (2016).
00350354
2016
j
05C15 05C10
neighbor sum distinguishing total coloring; planar graph; total coloring; discharging; combinatorial nullstellensatz
Summary: Let \(c\) be a proper total coloring of a graph \(G = (V, E)\) with integers \(1, 2, \ldots, k\). For any vertex \(v \in V(G)\), let \(\sum_c(v)\) denote the sum of colors of the edges incident with \(v\) and the color of \(v\). If for each edge \(uv \in E(G)\), \(\sum_c(u) \neq \sum_c(v)\), then such a total coloring is said to be neighbor sum distinguishing. The least \(k\) for which such a coloring of \(G\) exists is called the neighbor sum distinguishing total chromatic number and denoted by \(\chi_{\Sigma}^{\prime\prime}(G)\). \textit{M. Pil??niak} and \textit{M. Wo??niak} [Graphs Comb. 31, No. 3, 771--782 (2015; Zbl 1312.05054)] conjectured \(\chi_{\Sigma}^{\prime\prime}(G) \leq \Delta(G) + 3\) for any simple graph with maximum degree \(\Delta(G)\). It is known that this conjecture holds for any planar graph with \(\Delta(G) \geq 13\). In this paper, we prove that for any planar graph, \(\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 3, 14 \}\).
Zbl 1312.05054