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 χla(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 χla(G) = 2.
Related
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BibTeX
@misc{lau2026complete,
title = {{Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2}},
author = {Gee-Choon Lau and Wai Chee Shiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22808},
eprint = {2610.22808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22808},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2
AU - Gee-Choon Lau
AU - Wai Chee Shiu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.22808
UR - https://arxiv.org/abs/2610.22808
AB - 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.
ER -