Bipartite graphs are weak antimagic

TL;DR

This paper proves that all connected bipartite graphs except K2 have a weak antimagic labeling with repeated edge labels, and extends the concept to directed graphs, contributing to the Antimagic Graph Conjecture.

Contribution

It introduces a polynomial counting function for bipartite graphs and establishes the existence of weak antimagic labelings, extending the conjecture to directed and bidirected graphs.

Findings

Connected bipartite graphs (except K2) admit weak antimagic labelings.

The counting function for bipartite graphs is polynomial when labels are not distinct.

Extension of results to directed and bidirected graphs, proposing a directed version of the conjecture.

Abstract

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a given node are distinct. We study an associated counting function (replacing the upper bound on the possible labels by a variable) and prove that a variant of this counting function, when we do not require the labels to be distinct, is a polynomial if $G$ is bipartite. As a consequence, we show that every connected bipartite graph $G = (V, E)$ except $K_2$ admits a \emph{weakly} antimagic labeling, that is, each edge label is among $1, 2, ..., |E|$ (repetition allowed) and the sums of the labels on all edges incident with a given node are distinct. We also present a natural extension of these results to directed and bidirected graphs; this extension gives rise to a (bi-)directed version of the Antimagic Graph Conjecture, which might be of independent interest.