Adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least 10

Yulin Chang (), Qiancheng Ouyang () and Guanghui Wang ()
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Yulin Chang: Shandong University
Qiancheng Ouyang: Shandong University
Guanghui Wang: Shandong University

Journal of Combinatorial Optimization, 2019, vol. 38, issue 1, No 10, 185-196

Abstract: Abstract A (proper) total-k-coloring $$\phi :V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}$$ ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } is called adjacent vertex distinguishing if $$C_{\phi }(u)\ne C_{\phi }(v)$$ C ϕ ( u ) ≠ C ϕ ( v ) for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , where $$C_{\phi }(u)$$ C ϕ ( u ) is the set of the color of u and the colors of all edges incident with u. We use $$\chi ''_a(G)$$ χ a ′ ′ ( G ) to denote the smallest value k in such a coloring of G. Zhang et al. first introduced this coloring and conjectured that $$\chi ''_a(G)\le \Delta (G)+3$$ χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph G. For the list version of this coloring, it is known that $$ch''_a(G)\le \Delta (G)+3$$ c h a ′ ′ ( G ) ≤ Δ ( G ) + 3 for any planar graph with $$\Delta (G)\ge 11$$ Δ ( G ) ≥ 11 , where $$ch''_a(G)$$ c h a ′ ′ ( G ) is the adjacent vertex distinguishing total choosability. In this paper, we show that if G is a planar graph with $$\Delta (G)\ge 10$$ Δ ( G ) ≥ 10 , then $$ch''_a(G)\le \Delta (G)+3$$ c h a ′ ′ ( G ) ≤ Δ ( G ) + 3 .

Keywords: Adjacent vertex distinguishing total coloring; Planar graph; Combinatorial Nullstellensatz; Discharging (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10878-018-00375-w

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