Antimagic labeling of connected graphs

Prove that every connected graph other than the single edge graph K_2 is antimagic; that is, establish that each such graph admits an edge bijection whose induced vertex weights are all distinct.

Background

The paper recalls the classical antimagic-labeling conjecture, under which an edge labeling of a graph is required to induce pairwise distinct vertex weights. The conjecture concerns the existence of such a labeling for every connected graph except K_2, which is the known exceptional case.

This problem is included as one of the foundational unresolved questions motivating the discussion of antimagic and local antimagic labelings.

References

The famous unsolved problems are as follows. All connected graphs except $K_2$ are antimagic.

— A novel approach to determining chromatic number induced by labelings  (2608.17334 - Lau et al., 18 Aug 2026) in Section 1, Introduction

All trees except $K_2$ are antimagic.

— A novel approach to determining chromatic number induced by labelings  (2608.17334 - Lau et al., 18 Aug 2026) in Section 1, Introduction

Despite numerous results on this subject the conjecture is still open even for the class of trees.

— Antimagicness of join graphs  (2609.35245 - Beaudoire et al., 28 Sep 2026) in Section 1, Introduction

Since the class of cographs is the same as the class of $P_4$-free graphs, another open problem consists in proving the antimagicness for connected $P_k$-free graphs for $k\ge 5$.

— Antimagicness of join graphs  (2609.35245 - Beaudoire et al., 28 Sep 2026) in Section 3, Conclusion