Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2
Year: 2026
Type: preprint
Abstract: For a graph $G$, characterizing the structural behavior of achieving the minimal local antimagic chromatic number $χ_{la}(G) = 2$ 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 $χ_{la}(G) = 2$.
Source: arXiv (Cornell University)
Authors: Lau, Gee-Choon, Shiu, Wai Chee
Institution: Chinese University of Hong Kong
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Iowa State University
OpenAlex ID: W7222959247