Neighbor sum distinguishing total choosability of planar graphs

Cunquan Qu, Guanghui Wang (), Guiying Yan and Xiaowei Yu
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Cunquan Qu: Shandong University
Guanghui Wang: Shandong University
Guiying Yan: Chinese Academy of Sciences
Xiaowei Yu: Shandong University

Journal of Combinatorial Optimization, 2016, vol. 32, issue 3, No 16, 906-916

Abstract: Abstract A total-k-coloring of a graph G is a mapping $$c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}$$ c : V ( G ) ∪ E ( G ) → { 1 , 2 , ⋯ , k } such that any two adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) receive different colors. For a total-k-coloring of G, let $$\sum _c(v)$$ ∑ c ( v ) denote the total sum of colors of the edges incident with v and the color of v. If for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$\sum _c(u)\ne \sum _c(v)$$ ∑ c ( u ) ≠ ∑ 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 $$\chi _{\Sigma }^{''}(G)$$ χ Σ ′ ′ ( G ) . Pilśniak and Woźniak conjectured $$\chi _{\Sigma }^{''}(G)\le \Delta (G)+3$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph with maximum degree $$\Delta (G)$$ Δ ( G ) . In this paper, we prove that for any planar graph G with maximum degree $$\Delta (G)$$ Δ ( G ) , $$ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}$$ c h Σ ′ ′ ( G ) ≤ max { Δ ( G ) + 3 , 16 } , where $$ch^{''}_{\Sigma }(G)$$ c h Σ ′ ′ ( G ) is the neighbor sum distinguishing total choosability of G.

Keywords: Neighbour sum distinguishing total choosability; Planar graph; Total coloring; Discharging; Combinatorial Nullstellensatz; 05C15T (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10878-015-9911-9

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