On uniquely colorable Cayley graphs

Source 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.