Quasigroups defining Eulerian paths in complete graphs

This paper provides a graph theoretical contribution to the solution of a problem that was so far dealt with purely combinatorial methods: For which values of r does a P-quasigroup exist which defines an Eulerian path in the complete graph K_{2r + 1}? We start with a nearly linear factor F₀ whose edges are all of different “lengths” and obtain a rotative partition of K_{2r + 1} into nearly linear factors. At every vertex V_h, the 2r edges adjacent to V_h are thus partitioned into r transitions. Successive transitions so defined by F₀ are associated to edges of F₀ forming a sequence of the form (?i₀, i₁), (?i₁, i₂), (?i₂, i₃),…. If S is any closed path made up of these successive transitions, then S is described cyclically by a transition sequence of edges of F₀ of the form: I(S) = {(?i₀, i₁), (?i₁, i₂),…, (?i_t?1, i₀)}, used cyclically m times; S is Eulerian if and only if m = 2r + 1 and t = r. In particular, if t = r and G.C.D. (Σ_{j = 1}^ri_j, 2r + 1) = 1, then I(S) describes an Eulerian path. A numerical example is given, which solves the given problem whenever 2r + 10 (mod 7). Incidentally, Kotzig has shown that the problem has no solution when r = 2.