Every graph is uniform-span (2,2)-choosable: Beyond the 1-2 conjecture
Year: 2026
Type: article
Abstract: For a simple graph πΊ = ( π , πΈ ) , a proper total weighting is a mapping π€ : π βͺ πΈ β β such that for every edge π’ β’ π£ β πΈ , π€ β‘ ( π’ ) + β π β π’ π€ β‘ ( π ) β π€ β‘ ( π£ ) + β π β π£ π€ β‘ ( π ) . The graph G is ( 2 , 2 ) - choosable if, for any list assignment L that assigns to each z in π βͺ πΈ a set πΏ β‘ ( π§ ) of two real numbers, there exists a proper total weighting w with π€ β‘ ( π§ ) β πΏ β‘ ( π§ ) for every π§ β π βͺ πΈ . Wong and Zhu, and independently PrzybyΕo and WoΕΊniak conjectured that every simple graph is ( 2 , 2 ) -choosable. This conjecture remains open. For a set { π , π } β β , its span is | π β π | . We call G uniform-span ( 2 , 2 ) -choosable if for every list assignment L where all lists πΏ β‘ ( π§ ) have size two and have a common span, there exists a proper total weighting w with π€ β‘ ( π§ ) β πΏ β‘ ( π§ ) for all π§ β π βͺ πΈ . In this paper, we present a novel lemma and perform comprehensive enhancements to our previous algorithm. These contributions enable us to prove that every graph is uniform-span ( 2 , 2 ) -choosable, and thus admits a proper total weighting using any prescribed pair { π , π } β β (including { 1 , 2 } ). This confirms and extends the 1-2 conjecture.
Authors: Kecai Deng, Hongyuan Qiu
Institutions: Huaqiao University, Xiamen University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,150
OpenAlex ID: W6966985763
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