The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

Lin Sun, Xiaohan Cheng and Jianliang Wu ()
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Lin Sun: Changji University
Xiaohan Cheng: Shandong University
Jianliang Wu: Shandong University

Journal of Combinatorial Optimization, 2017, vol. 33, issue 2, No 23, 779-790

Abstract: Abstract A total [k]-coloring of a graph G is a mapping $$\phi $$ ϕ : $$V(G)\cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$ V ( G ) ∪ E ( G ) → [ k ] = { 1 , 2 , … , k } such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) receive the same color. In a total [k]-coloring $$\phi $$ ϕ of G, let $$C_{\phi }(v)$$ C ϕ ( v ) denote the set of colors of the edges incident to v and the color of v. If for each edge uv, $$C_{\phi }(u)\ne C_{\phi }(v)$$ C ϕ ( u ) ≠ C ϕ ( v ) , we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. $$\chi ''_{a}(G)$$ χ a ′ ′ ( G ) denotes the smallest value k in such a coloring of G. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph G with maximum degree $$\Delta \ge 8$$ Δ ≥ 8 contains no adjacent 4-cycles, then $$\chi ''_{a}(G)\le \Delta +3$$ χ a ′ ′ ( G ) ≤ Δ + 3 .

Keywords: Planar graph; Adjacent vertex distinguishing total coloring; Combinatorial Nullstellensatz; Discharging method; 05C15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-016-0004-1

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