Uniquely colorable Cayley graphs

Abstract

It is shown that the chromatic number χ ( G ) = k of a uniquely colorable Cayley graph G over a group Γ is a divisor of ∣Γ ∣ = n . Each color class in a k -coloring of G is a coset of a subgroup of order n / k of Γ . Moreover, it is proved that ( k − 1) n is a sharp lower bound for the number of edges of a uniquely k -colorable, noncomplete Cayley graph over an abelian group of order n . Finally, we present constructions of uniquely colorable Cayley graphs by graph products.