Paarweise kantendisjunkte maximale matchings in normalen periodischen Pflasterungen. (On pairwise edge-disjoint maximal matchings in normal periodic tilings) (Q1121531)

Let P be a normal periodic tiling of the plane, and let G be the graph of P. (The vertices of G are the points of the plane, each belonging to at least three tiles, and the edges of G are the arcs, each forming the intersection of exactly two tiles.) A matching M of G is a family of disjoint edges. M is said to be maximal if M is not a proper subset of another matching of G. Let m(G) denote the maximum number of pairwise edge-disjoint maximal matchings of G. The following theorem is proved: If P is a normal periodic tiling then m(P)\(\leq 13\). Equality holds if and only if P is combinatorially equivalent to the Laves-tiling \([3.12^ 2]\).