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

TL;DR

The paper addresses antimagic labeling for connected $(k,2)$-bipartite biregular graphs, building on prior results for $(s,t)$-bipartite graphs with $s\ge t+2$ and one odd. It introduces a layered, parity-controlled edge-labelling framework that starts from a root vertex of degree $k$, partitions $V$ into distance layers $V_i$ and edge layers $L_i$, and carefully assigns labels to ensure distinct vertex-sums $\sigma(u)$. For odd $k\ge 3$, the construction yields antimagic labellings for all connected $(k,2)$-bipartite graphs (and extends to non-connected graphs); for even $k\ge 4$, the method is adapted to obtain antimagic labellings for connected graphs as well. The approach uses a three-step process per layer, including a subset $F_i$ of edges to control partial sums, followed by completing the labelling to enforce parity constraints and resolve potential conflicts, providing a constructive resolution to antimagicness for large families of $(k,2)$-bipartite graphs. Overall, the work advances the antimagic labeling program by resolving broad classes of biregular bipartite graphs and offering a scalable, constructive framework.

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.