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We deal with a family of undirected Cayley graphs \(X_n\) which are unions of disjoint Hamilton cycles, and some of their properties, where \(n\) runs over the positive integers. It is proved that \(X_n\) is a bipartite graph when \(n\) is even. If \(n\) is an odd number, we count the number of different colored triangles in \(X_n\).