arXiv subjects

Kecai Deng

Publications and source records attributed to Kecai Deng.

3 recordsLinked to original sources

arXiv2026

Regular graphs are universally 3-edge-weightable

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular graph has a proper $\{a,b,c\}$-edge weighting. We prove that this remains valid for all 3-element set $\{a,b,c\}$ with $c-b \neq b-a$. Consequently, every nice regular graph is universally $3$-edge-weightable.

math.CO↗

arXiv2025

Bipartite graphs with minimum degree at least 15 are antimagic

An antimagic {labeling} of a graph $G=(V,E)$ is a one-to-one mapping $f: E\rightarrow\{1,2,\ldots,|E|\}$, ensuring that the vertex sums in $V$ are pairwise distinct, where a vertex sum of a vertex $v$ is defined as the sum of the labels of the edges incident to $v$. A graph is called antimagic if it admits an antimagic labeling. The Antimagic Labeling Conjecture, proposed by Hartsfield and Ringel in 1990, posits that every connected graph other than $K_2$ is antimagic. The conjecture was confirmed for graphs of average degree at least 4,182 in 2016 by Eccles, where it was stated that a similar approach could not reduce the bound below 1,000 from 4,182. This paper shows that every bipartite graph with minimum degree at least 15 is antimagic. Our approach relies on three tools: a consequence of K\"{o}nig's Theorem, the existence of a subgraph of a specific size that avoids Eulerian components, and a labeling lemma that ensures some vertex sums are divisible by three while others are not.

math.CO↗

arXiv2025

Every graph is uniform-span $(2,2)$-choosable: Beyond the 1-2 conjecture

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 Przyby{\l}o and Wo\'{z}niak 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.

math.CO↗