Edge proximity and matching extension in planar triangulations

Let G be a graph with at least 2(m + n + 1) vertices. Then G is E(m, n) if for each pair of disjoint matchings M,N ⊆ E(G) ofsizemand n respectively, there exists a perfect matching F in G such that M ⊆ F and F ∩ N = ∅. In the present paper we wish to study property E(m, n) for the various values of integers m and n when the graphs in question are restricted to be planar. It is known that no planar graph is E(3, 0) or E(2, 1). In this paper we show that in planar even triangulations, matchings of size three satisfying certain proximity conditions can be extended to perfect matchings. We also determine precisely for which values of m and n, the property E(m, n) holds when the graphs involved are even triangulations or near-triangulations of the plane