Compatible Euler tours of

Let Kn∗∗ be the complete symmetric directed graph on n vertices with a directed loop at each vertex. Barth, Bond and Raspaud conjectured that there exist n − 1 pairwise compatible Euler tours of Kn∗∗. In this paper we construct a set of 2k pairwise compatible Euler tours of K2k + 1∗∗ for all k, thereby verifying half of their conjecture. We also improve their lower bound for number of pairwise compatible Euler tours of K2k∗∗ to 2k − 2.