Every graph is (2,3)-choosable

Abstract

A total weighting of a graph G is a mapping ϕ that assigns to each element zV (G)∪E(G) a weight ϕ(z). A total weighting ϕ is proper if for any two adjacent vertices u and v, ∑ eE(u) ϕ(e)+ϕ(u)≠∑ eE(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 zV (G)∪E(G).

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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