an:07148550
Zbl 1430.05044
Zhao, Xue; Xu, Chang-Qing
Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles
EN
Discuss. Math., Graph Theory 40, No. 1, 243-253 (2020).
00443686
2020
j
05C15
neighbor sum distinguishing total coloring; discharging method; planar graph
Summary: For a given graph \(G = (V (G), E(G))\), a proper total coloring \(\varphi : V (G) \cup E(G) \rightarrow \{1, 2, \dots, k\}\) is neighbor sum distinguishing if \(f(u) \neq f(v)\) for each edge \(uv \in E(G)\), where \(f(v) = \Sigma_{ uv \in E(G)} \varphi (uv)+ \varphi (v)\), \(v \in V (G)\). The smallest integer \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, 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)] first introduced this coloring and conjectured that \(\chi^{\prime\prime}_\Sigma(G)\le \Delta ( G ) + 3\) for any graph with maximum degree \(\Delta (G)\). In this paper, by using the discharging method, we prove that for any planar graph \(G\) without 5-cycles, \(\chi^{\prime\prime}_\Sigma ( G ) \le \max \left\{ \Delta ( G ) + 2,10\right\}\). The bound \(\Delta (G) + 2\) is sharp. Furthermore, we get the exact value of \(\chi^{\prime\prime}_\Sigma( G )\) if \(\Delta (G) \geq 9\).
Zbl 1312.05054