Abstract
In [4] it was shown that in a 5-connected even planar triangulationG, every matchingMof size|M|<|V(G)|/2can be extended to a perfect matching ofG, as long as the edges ofMlie at distance at least 5 from each other. Somewhat later in [7], the following result was proved. LetGbe a 5-connected triangulation of a surface sigma different from the sphere. Let chi=chi(sigma)be the Euler characteristic of sigma. SupposeV0 subset of V(G)with|V(G)-V0|even andMandNare two matchings inG-V0such thatM boolean AND N= null . Further suppose that the pairwise distance between two elements ofV0?M?Nis at least 5 and the face-width of the embedding ofGin sigma is at leastmax{20|M|-8 chi-23,6}. Then there is a perfect matchingM0inG-V0which containsMsuch thatM0 boolean AND N= null . In the present paper, we present some results which, in a sense, lie in the gap between the two above theorems, in that they deal with restricted matching extension in a planar triangulation when a set of vertices which lie pairwise at sufficient distance from one another has been deleted. In particular, we prove a planar analogue of the result in [7] stated above.