Isomorphism problem for relational structures with a cyclic automorphism
European Journal of Combinatorics
Multipliers and generalized multipliers of cyclic objects and cyclic codes
Journal of Combinatorial Theory Series A
A non-Cayley-invariant Cayley graph of the elementary Abelian group of order 64
Discrete Mathematics
A´da´m's conjecture is true in the square-free case
Journal of Combinatorial Theory Series A
Isomorphism problem for Cayley graphs of Z3p
Discrete Mathematics
On Ádám's conjecture for circulant graphs
Discrete Mathematics
Isomorphisms of finite Cayley digaphs of bounded valency
Journal of Combinatorial Theory Series B
On the isomorphism problem for cyclic combinatorial objects
Discrete Mathematics
Transitive Permutation Groups of Prime-Squared Degree
Journal of Algebraic Combinatorics: An International Journal
On isomorphisms of finite Cayley graphs: a survey
Discrete Mathematics
On the Cayley isomorphism problem
Discrete Mathematics
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A ternary relational structure X is an ordered pair (V,E), where E ⊂ V3. A ternary relational structure X is a Cayley ternary relational structure of a group G if the left regular representation of G is contained in the automorphism group of X. A group G is a CI-group with respect to ternary relational structures if whenever X and X' are isomorphic Cayley ternary relational structures of G, then X and X' are isomorphic if and only if they are isomorphic by an automorphism of G. In this paper, we will provide a (relatively short) list of all possible CI-groups with respect to color ternary relational structures. All of these groups have order 2dn, 0≤d≤ 5, and n a positive integer with gcd(n, φ(n)) = 1, where φ is Euler's phi function. If d = 0, it has been shown by Pálfy that Zn is a CI-group with respect to every class of combinatorial objects. We then show that of the possible CI-groups with respect to color ternary relational structures of order 2n and 4n with a cyclic Sylow 2-subgroup, most are CI-groups with respect to ternary relational structures, and in the unresolved cases, give a necessary and sufficient condition for the group to be a CI-group with respect to ternary relational structures. In particular, we determine all cyclic CI-groups with respect to ternary relational structures. Finally, a group G that is a CI-group with respect to ternary relational structures is also a CI-group with respect to binary relational structures (i.e. graphs and digraphs). Some of the groups considered in this paper are not known to be CI-groups with respect to graphs or digraphs, and we thus provide new examples of CI-groups with respect to graphs and digraphs.