2025/06/17 by Deng, Kecai, Qiu, Hongyuan
#05C15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.14253
For a simple graph G=(V,E), a proper total weighting is a mapping w: V∪ E→ \mathbb R such that for every edge uv∈ E, w(u)+∑e\ni uw(e)≠ w(v)+∑e\ni vw(e). The graph G is said (2,2)-choosable if, for any list assignment L that assigns to each z in V∪ E a set L(z) of two real numbers, there exists a proper total weighting w with w(z)∈ L(z) for every z∈ V∪ E. Wong and Zhu, and independently Przybyło and Woźniak conjectured that every simple graph is (2,2)-choosable. This conjecture remains open. For a set \a,b\⊂ \mathbb R, its span is defined as |b-a|. We call a graph G=(V,E) uniform-span (2,2)-choosable if, for any list assignment L that assigns to every z∈ V∪ E a two-element list of a common span, there exists a proper total weighting respect to the assignment. 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. This confirms the 1-2 conjecture in full generality, and provides supporting evidence for the (2,2)-choosable conjecture.