Euler tours and a game with dominoes Original Research Article

Let Knl denote the complete graph with vertices v0,v1,…,vn-1, together with the loops e1 = vivi for i = 0, 1,…, n − 1. Suppose n3 mod 4 and that the edges of Knl are coloured either blue or red in such a way that each colour class has the same number of edges. We show that there is an ordering f1, f2,…,fm of the edges of Kn′, alternating colours, such that for i = 1, 2,…, m − 1, the edges f1, f2,…, fi form a trail Ti of Knl with fi+1 incident with at least one end of Ti. This solves a problem proposed by González-Acuña and Urrutia (1985).