On restricted matching extension in planar graphs

Let G be a connected graph with at least 2(m + n + 1) vertices. Then G is E(m, n) if for each pair of disjoint matchings M, N subset of or equal to E(G) of size m and n, respectively, there exists a perfect matching F in G such that MCF and F boolean ANDN = 0. In the present paper, we wish to study the 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 (Plummer, Annals of Discrete Mathematics 41 (1989) 347-354) that no planar graph is E(3.0). This result is improved in the present paper by showing that no planar graph is E(2, I). This severely limits the values of m and n for which a planar graph can have property E(m,n) and leads us to consider the properties E(1, n) and E(0, n) for certain classes of planar graphs. Sharpness of the various results is also explored. (C) 2001 Elsevier Science B.V. All rights reserved.