关于唯一可着色的Cayley图
On uniquely colorable Cayley graphs
- University of Niš（尼什大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究解决了Klotz和Sander提出的两个关于唯一可着色Cayley图的开放问题，构造了无限族唯一3-可着色整循环图并否定相关猜想，还建立通用代数构造证明非阿贝尔群上存在符合要求的唯一可着色Cayley图。
中文摘要 AI 辅助
我们解决了Klotz和Sander（2017）提出的关于唯一可着色Cayley图的两个开放问题。首先，我们构造了一个无限族的唯一3-可着色整循环图，其团数为2。这对问题3.6给出了否定回答，该问题询问是否每个唯一可着色的循环图都满足χ(G)=ω(G)。由于验证唯一可着色性本质上依赖于精确的独立数，我们证明传统的谱界无法紧密捕捉该参数，因此需要基于精确结构同构的严格组合证明。其次，我们建立了一个通用代数构造，证明在非阿贝尔群上存在唯一可着色的Cayley图，其颜色类是严格不同子群的左陪集。通过利用非正规子群的右陪集划分，该结果对问题2.4给出了明确的肯定回答。
英文摘要
We resolve two open problems regarding uniquely colorable Cayley graphs posed by Klotz and Sander (2017). First, we construct an infinite family of uniquely 3-colorable integral circulant graphs with a clique number of 2. This provides a negative answer to Problem 3.6, which asks whether every uniquely colorable circulant graph satisfies $χ(G) = ω(G)$. Because verifying unique colorability inherently relies on the exact independence number, we demonstrate that traditional spectral bounds fail to tightly capture this parameter, necessitating a rigorous combinatorial proof based on exact structural isomorphisms. Second, we establish a general algebraic construction proving the existence of uniquely colorable Cayley graphs over nonabelian groups whose color classes are left cosets of strictly distinct subgroups. By utilizing right-coset partitions of non-normal subgroups, this result provides a definitive affirmative answer to Problem 2.4.
