Every graph is uniform-span $(2,2)$-choosable: Beyond the 1-2 conjecture
Kecai Deng, Hongyuan Qiu
TL;DR
The paper addresses the uniform-span $(2,2)$-choosability and its implications for the 1-2 conjecture by introducing a novel lemma and algorithmic improvements. It provides a constructive proof that every graph admits a proper $\{0,a\}$-total weighting for any $a>0$, using a layered independent-set decomposition and well-subgraph techniques to propagate weight. Consequently, it establishes that every graph is $uniform\text{-}span(2,2)$-choosable, thereby resolving the 1-2 conjecture in full generality and supporting the $(2,2)$-choosable conjecture. The results offer a general framework for uniform-span choosability and suggest avenues for extending these methods to related total-weighting and list-choosability problems.
Abstract
For a simple graph $G=(V,E)$, a \emph{proper total weighting} is a mapping $w: V\cup E\rightarrow \mathbb R$ such that for every edge $uv\in E$, $w(u)+\sum_{e\ni u}w(e)\neq w(v)+\sum_{e\ni v}w(e)$. The graph $G$ is said $(2,2)$-\emph{choosable} if, for any list assignment $L$ that assigns to each $z$ in $V\cup E$ a set $L(z)$ of two real numbers, there exists a {proper total weighting} $w$ with $w(z)\in L(z)$ for every $z\in V\cup E$. Wong and Zhu, and independently Przybyło and Woźniak conjectured that every simple graph is $(2,2)$-choosable. This conjecture remains open. For a set $\{a,b\}\subset \mathbb R$, its span is defined as $|b-a|$. We call a graph $G=(V,E)$ \emph{uniform-span} $(2,2)$-\emph{choosable} if, for any list assignment $L$ that assigns to every $z\in V\cup E$ 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.