A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Jihui Wang (), Qiaoling Ma, Xue Han and Xiuyun Wang
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Jihui Wang: University of Jinan
Qiaoling Ma: University of Jinan
Xue Han: University of Jinan
Xiuyun Wang: University of Jinan

Journal of Combinatorial Optimization, 2016, vol. 32, issue 2, No 17, 626-638

Abstract: Abstract Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$\phi $$ ϕ be a total $$k$$ k -coloring of $$G$$ G using the color set $$\{1,\ldots , k\}$$ { 1 , … , k } . Let $$\sum _\phi (u)$$ ∑ ϕ ( u ) denote the sum of the color of the vertex $$u$$ u and the colors of all incident edges of $$u$$ u . A $$k$$ k -neighbor sum distinguishing total coloring of $$G$$ G is a total $$k$$ k -coloring of $$G$$ G such that for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$\sum _\phi (u)\ne \sum _\phi (v)$$ ∑ ϕ ( u ) ≠ ∑ ϕ ( v ) . By $$\chi ^{''}_{nsd}(G)$$ χ n s d ′ ′ ( G ) , we denote the smallest value $$k$$ k in such a coloring of $$G$$ G . Pilśniak and Woźniak first introduced this coloring and conjectured that $$\chi _{nsd}^{''}(G)\le \Delta (G)+3$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph $$G$$ G . In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with $$\Delta (G)\ge 7$$ Δ ( G ) ≥ 7 . Moreover, we also show that $$\chi _{nsd}^{''}(G)\le \Delta (G)+2$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 2 for planar graphs without intersecting triangles with $$\Delta (G) \ge 9$$ Δ ( G ) ≥ 9 . Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

Keywords: Neighbor sum distinguishing total coloring; Combinatorial Nullstellensatz; Intersecting triangles; Planar graph; 05C15 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-015-9886-6

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