Proximity Thresholds for Matching Extension in Planar and Projective Planar Triangulations

A graph on at least 2(m + 1) vertices with a perfect matching is said to be m-extendable if, given any matching M with vertical bar M vertical bar = m, there is a perfect matching F in G such that M subset of F. It has been known for some time that no planar graph is 3-extendable. More recently, a graph on at least 2m + 2 vertices has been defined to be distance d m-extendable if given any matching M with vertical bar M vertical bar = m in which the edges lie at pair-wise distance at least d, there is a perfect matching containing M. In another recent article it was shown that a 5-connected even planar triangulation is distance 2 3-extendable, but not necessarily distance 2 4-extendable. Moreover, it has also been shown that such a graph need not be distance 3 10-extendable. Hence it is of interest to know the largest integer m such that distance d m-extendable holds for all such graphs. The above tells us that for d = 2, the maximum value is m = 3. In the present work, it is shown that for d = 5, in fact, there is no upper bound on m such that a 5-connected, even planar, or projective planar, graph G on at least 2m + 2 vertices is distance 5 m-extendable. (C) 2011 Wiley Periodicals, Inc. J Graph Theory 67: 38-46, 2011