Every graph is (2,3)-choosable

Combinatorica volume 36, pages121–127(2016)Cite this article

Abstract

A total weighting of a graph G is a mapping ϕ that assigns to each element z ∈ V (G)∪E(G) a weight ϕ(z). A total weighting ϕ is proper if for any two adjacent vertices u and v, ∑ e∈E(u) ϕ(e)+ϕ(u)≠∑ e∈E(v) ϕ(e)+ϕ(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ϕ with ϕ(z) ∈ L(z) for each element z ∈ V (G)∪E(G).

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References

  1. [1]

    L. Addario-Berry, R. E. L. Aldred, K. Dalal and B. A. Reed: Vertex colouring edge partitions, J. Combin. Theory Ser. B 94 (2005), 237–244.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    L. Addario-Berry, K. Dalal, C. McDiarmid, B. A. Reed and A. Thomason: Vertex-colouring edge-weightings, Combinatorica 27 (2007), 1–12.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    N. Alon: Combinatorial Nullstellensatz, Combin. Prob. Comput. 8 (1999), 7–29.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    N. Alon and M. Tarsi: A nowhere zero point in linear mappings, Combinatorica 9 (1989), 393–395.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    N. Alon and M. Tarsi: Colorings and orientations of graphs, Combinatorica 12 (1992), 125–134.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    T. Bartnicki, J. Grytczuk and S. Niwczyk: Weight choosability of graphs, J. Graph Theory 60 (2009), 242–256.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    M. Karoński, T. Luczak and A. Thomason: Edge weights and vertex colours, J. Combin. Theory Ser. B 91 (2004), 151–157.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    M. Kalkowski: On 1,2-conjecture, manuscript, 20–8.

  9. [9]

    M. Kalkowski, M. Karoński and F. Pfender: Vertex-coloring edge-weightings: towards the 1-2-3- Conjecture, J. Combin. Theory Ser. B 100 (2010), 347–349.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    J. Przybyło and M. Woźniak: On a 1-2 conjecture, Discrete Mathematics and Theoretical Computer Science 12 (2010), 101–108.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    J. Przybyło and M. Woźniak: Total weight choosability of graphs, Electronic J. Combinatorics 18 (2011), no. 1, Paper #P112.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    T. Wang and Q. L. Yu: A note on vertex-coloring 13-edge-weighting, Frontier Math. 4 in China 3 (2008), 1–7.

    Article  Google Scholar 

  13. [13]

    T. Wong, J. Wu and X. Zhu: Total weight choosability of Cartesian product of graphs, European J. Combinatorics 33 (2012), 1725–1738.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    T. Wong, D. Yang and X. Zhu: List total weighting of graphs, in: Fete of combinatorics and computer science, 337–353. Bolyai Soc. Math. Stud., 20, János Bolyai Math. Soc., Budapest, 2010.

    Google Scholar 

  15. [15]

    T. Wong and X. Zhu: Total weight choosability of graphs, J. Graph Theory 66 (2011), 198–212.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    T. Wong and X. Zhu: Permanent index of matrices associated with graphs, manuscript.

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Author information

Affiliations

  1. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Tsai-Lien Wong

  2. National Center for Theoretical Sciences, Hsinchu, Taiwan

    Tsai-Lien Wong

  3. Department of Mathematics, Zhejiang Normal University, Zhejiang, China

    Xuding Zhu

Authors
  1. Tsai-Lien Wong
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  2. Xuding Zhu
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Corresponding author

Correspondence to Xuding Zhu.

Additional information

Grant numbers: NSC102-2115-M-110-006-MY2.

Grant numbers: NSF11171310 and ZJNSF Z6110786.

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Wong, TL., Zhu, X. Every graph is (2,3)-choosable. Combinatorica 36, 121–127 (2016). https://doi.org/10.1007/s00493-014-3057-8

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Mathematics Subject Classification (2000)

  • 05C50
  • 05C78