Antimagic labellings of (k, 2)-bipartite biregular graphs

TL;DR

This paper proves that certain bipartite biregular graphs with degree 2 are antimagic, extending previous results to new classes of graphs with specific degree conditions.

Contribution

It extends the class of bipartite biregular graphs known to be antimagic to include (k, 2)-bipartite graphs for both even and odd k.

Findings

Connected (k, 2)-bipartite biregular graphs with k ≥ 4 even are antimagic.

Connected (k, 2)-bipartite biregular graphs with k ≥ 3 odd are antimagic.

The result generalizes previous antimagic graph theorems to new bipartite biregular cases.

Abstract

An antimagic labelling of a graph is a bijection from the set of edges to $\{1, 2, \ldots , m\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of a vertex is the sum of labels on the edges incident to it. We say a graph is antimagic if it has an antimagic labelling. In 2023, it has been proven that connected $(k, l)$-bipartite graphs are antimagic if $k \geq l + 2$ and one of k or l is odd. In this paper, we extend this result to connected $(k, 2)$-bipartite biregular graphs for $k \geq 4$ even, and to $(k, 2)$-bipartite biregular graphs for $k \geq 3$ odd.