Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2

Abstract

For a graph $G$, characterizing the structural behavior of achieving the minimal local antimagic chromatic number ${\chi }_{la}(G)=2$ remains a fundamental open problem in the study of vertex-distinguishing edge labelings. This paper establishes complete characterizations for a significant structural class of disconnected graphs, specifically addressing disconnected one-point unions of cycles with at most three components. By introducing a robust algebraic approach, we show that these specific graph architectures form distinct, manageable solution families under a local antimagic 2-coloring framework. Crucially, our structural framework provides the theoretical basis for extending these complete characterizations to an arbitrary number of components, demonstrating unique edge assignments up to isomorphism. These results offer key insights into the structural properties governing low-chromatic antimagic invariants and offer significant progress toward a unified classification of graphs with ${\chi }_{la}(G)=2$.

arXiv commentWe revised the Abstract and Introduction sections to better articulate the manuscript's core contribution to the classification and characterization of bipartite graphs with local antimagic chromatic number 2. The Conclusion and Open Problems section has been refined accordingly