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Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz

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Abstract

Let G = (V,E) be a graph and φ be a total coloring of G by using the color set {1, 2, ...,k}. Let f(ν) denote the sum of the color of the vertex ν and the colors of all incident edges of ν. We say that φ is neighbor sum distinguishing if for each edge uν ∈ E(G), f(u) ≠ f(ν). The smallest number k is called the neighbor sum distinguishing total chromatic number, denoted by χ nsd″(G). Pilśniak and Woźniak conjectured that for any graph G with at least two vertices, χ nsd″(G) ⩾ Δ(G) + 3. In this paper, by using the famous Combinatorial Nullstellensatz, we show that χ nsd″(G) ⩾ 2Δ(G)+col(G)−1, where col(G) is the coloring number of G. Moreover, we prove this assertion in its list version.

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References

  1. Alon N. Combinatorial Nullstellensatz. Combin Probab Comput, 1999, 8: 7–29

    Article  MATH  MathSciNet  Google Scholar 

  2. Bondy J A, Murty U S R. Graph Theory with Applications. New York: North-Holland, 1976

    MATH  Google Scholar 

  3. Chen X E. On the adjacent vertex distinguishing total coloring numbers of graphs with Δ = 3. Discrete Math, 2008, 308: 4003–4007

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheng X H, Wu J L, Huang D J, et al. Neighbor sum distinguishing total colorings of planar graphs with maximum degree Δ. Submitted, 2013

    Google Scholar 

  5. Dong A J, Wang G H. Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree. Acta Math Sin Engl Ser, in press, 2013

    Google Scholar 

  6. Huang D J, Wang W F. Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree (in Chinese). Sci Sin Math, 2012, 42: 151–164

    Article  Google Scholar 

  7. Huang P Y, Wong T L, Zhu X D. Weighted-1-antimagic graphs of prime power order. Discrete Math, 2012, 312: 2162–2169

    Article  MATH  MathSciNet  Google Scholar 

  8. Kalkowski M, Karoński M, Pfender F. Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture. J Combin Theory Ser B, 2010, 100: 347–349

    Article  MATH  MathSciNet  Google Scholar 

  9. Karoński M, Łuczak T, Thomason A. Edge weights and vertex colors. J Combin Theory Ser B, 2004, 91: 151–157

    Article  MATH  MathSciNet  Google Scholar 

  10. Li H L, Ding L H, Liu B Q, et al. Neighbor sum distinguishing total colorings of planar graphs. J Combin Optim, 2013, doi: 10.1007/s10878-013-9660-6

    Google Scholar 

  11. Li H L, Liu B Q, Wang G H. Neighbor sum distinguishing total colorings of K4-minor free graphs. Front Math China, 2013, 8: 1351–1366

    Article  MathSciNet  Google Scholar 

  12. Pilśniak M, Woźniak M. On the adjacent-vertex-distinguishing index by sums in total proper colorings. Preprint, http://www.ii.uj.edu.pl/preMD/index.php

  13. Przybyło J. Irregularity strength of regular graphs. Electronic J Combin, 2008, 15: #R82

    Google Scholar 

  14. Przybyło J. Linear bound on the irregularity strength and the total vertex irregularity strength of graphs. SIAM J Discrete Math, 2009, 23: 511–516

    Article  Google Scholar 

  15. Przybyło J. Neighbour distinguishing edge colorings via the Combinatorial Nullstellensatz. SIAM J Discrete Math, 2013, 27: 1313–1322

    Article  MATH  MathSciNet  Google Scholar 

  16. Przybyło J, Woźniak M. Total weight choosability of graphs. Electronic J Combin, 2011, 18: #P112

    Google Scholar 

  17. Przybyło J, Woźniak M. On a 1, 2 conjecture. Discrete Math Theor Comput Sci, 2010, 12: 101–108

    MATH  MathSciNet  Google Scholar 

  18. Scheim E. The number of edge 3-coloring of a planar cubic graph as a permanent. Discrete Math, 1974, 8: 377–382

    Article  MATH  MathSciNet  Google Scholar 

  19. Seamone B. The 1-2-3 conjecture and related problems: A survey. ArXiv:1211.5122, 2012

    Google Scholar 

  20. Wang W F, Huang D J. The adjacent vertex distinguishing total coloring of planar graphs. J Combin Optim, 2012, doi: 10.1007/s10878-012-9527-2

    Google Scholar 

  21. Wang W F, Wang P. On adjacent-vertex-distinguishing total coloring of K 4-minor free graphs (in Chinese). Sci China Ser A, 2009, 39: 1462–1472

    Google Scholar 

  22. Wang Y Q, Wang W F. Adjacent vertex distinguishing total colorings of outerplanar graphs. J Combin Optim, 2010, 19: 123–133

    Article  MATH  Google Scholar 

  23. Wong T L, Zhu X D. Total weight choosability of graphs. J Graph Theory, 2011, 66: 198–212

    Article  MATH  MathSciNet  Google Scholar 

  24. Wong T L, Zhu X D. Antimagic labelling of vertex weighted graphs. J Graph Theory, 2012, 3: 348–359

    Article  MathSciNet  Google Scholar 

  25. Zhang Z F, Chen X E, Li J W, et al. On adjacent-vertex-distinguishing total coloring of graphs. Sci China Ser A, 2005, 48: 289–299

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, China

    LaiHao Ding & GuangHui Wang

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 10080, China

    GuiYing Yan

Authors
  1. LaiHao Ding
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  2. GuangHui Wang
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  3. GuiYing Yan
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Correspondence to GuangHui Wang.

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Ding, L., Wang, G. & Yan, G. Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz. Sci. China Math. 57, 1875–1882 (2014). https://doi.org/10.1007/s11425-014-4796-0

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  • DOI: https://doi.org/10.1007/s11425-014-4796-0

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