Cycles containing matchings and pairwise compatible euler tours

Let M be a matching in a graph G such that d(x) + d(y) ≥ |G| for all pairs of independent vertices x and y of G that are incident with M. We determine a necessary and sufficient condition for M to be contained in a cycle of G. This extends results of Häggkvist and Berman, and implies that if M is a 1‐factor of G and |G|  0 (mod 4), then M is contained in a Hamilton cycle of G. We use our results to deduce that an eulerian graph of minimum degree 2k contains k pairwise compatible Euler tours.