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Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

Author

Listed:
  • Hongjie Song

    (Hebei University of Technology
    Hebei Province Key Laboratory of Big Data Calculation)

  • Changqing Xu

    (Hebei University of Technology
    Hebei Province Key Laboratory of Big Data Calculation)

Abstract

Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}$$ ϕ : V ∪ E → { 1 , 2 , … , k } be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring $$\phi $$ ϕ is neighbor sum distinguishing if $$f(u)\ne f(v)$$ f ( u ) ≠ f ( v ) for each edge $$uv\in E(G)$$ u v ∈ E ( G ) . The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by $$\chi _{\Sigma }''(G)$$ χ Σ ′ ′ ( G ) . Pilśniak and Woźniak conjectured that $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that $$\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}$$ χ Σ ′ ′ ( G ) ≤ max { Δ ( G ) + 2 , 10 } for planar graph G without 4-cycles. The bound $$\Delta (G)+2$$ Δ ( G ) + 2 is sharp if $$\Delta (G)\ge 8$$ Δ ( G ) ≥ 8 .

Suggested Citation

  • Hongjie Song & Changqing Xu, 2017. "Neighbor sum distinguishing total coloring of planar graphs without 4-cycles," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1147-1158, November.
  • Handle: RePEc:spr:jcomop:v:34:y:2017:i:4:d:10.1007_s10878-017-0137-x
    DOI: 10.1007/s10878-017-0137-x
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    References listed on IDEAS

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    1. Jihui Wang & Qiaoling Ma & Xue Han & Xiuyun Wang, 2016. "A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 626-638, August.
    2. Cunquan Qu & Guanghui Wang & Guiying Yan & Xiaowei Yu, 2016. "Neighbor sum distinguishing total choosability of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 906-916, October.
    3. Hualong Li & Laihao Ding & Bingqiang Liu & Guanghui Wang, 2015. "Neighbor sum distinguishing total colorings of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 675-688, October.
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    Cited by:

    1. Xu, Changqing & Li, Jianguo & Ge, Shan, 2018. "Neighbor sum distinguishing total chromatic number of planar graphs," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 189-196.

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