List-antimagic labeling of vertex-weighted graphs

TL;DR

This paper introduces new antimagic labeling concepts for vertex-weighted and oriented graphs, proving bounds on the number of labels needed for such labelings to exist in broad classes of graphs.

Contribution

It establishes the existence of weighted-list-antimagic and oriented-antimagic labelings with specific bounds for graphs without isolated vertices or edges.

Findings

Graphs with no isolated vertices or edges are $rac{4n}{3}$-weighted-list-antimagic.

Graphs without isolated vertices admit $rac{2n}{3}$-oriented-antimagic orientations.

The results extend antimagic labelings to vertex-weighted and oriented graph classes.

Abstract

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.