Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Bašić, Milan
Institution: University of Nis
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6829897439
OpenAlex ID: W7208766586