## Zhanar Berikkyzy ; Axel Brandt ; Sogol Jahanbekam ; Victor Larsen ; Danny Rorabaugh - List-antimagic labeling of vertex-weighted graphs

dmtcs:5631 - Discrete Mathematics & Theoretical Computer Science, December 2, 2021, vol. 23, no. 3 - https://doi.org/10.46298/dmtcs.5631
List-antimagic labeling of vertex-weighted graphs

Authors: Zhanar Berikkyzy ; Axel Brandt ; Sogol Jahanbekam ; Victor Larsen ; Danny Rorabaugh

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.

Volume: vol. 23, no. 3
Section: Graph Theory
Published on: December 2, 2021
Accepted on: July 19, 2021
Submitted on: July 17, 2019
Keywords: Mathematics - Combinatorics

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