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A Theorem on Matchings in the Plane. 2. Some Planar Considerations

Abstract : Let G be a graph with (V(G)) = p points and (E(G)) = q lines. A matching in G is any set of lines in E(G) no two of which are adjacent. Matching M in G is said to be a perfect matching, or p.m., if every point of G is covered by a line of M. Let G be any graph with a perfect matching and suppose positive integer n or minus (p - 2)/2. Then G is n extendable if every matching in G containing n lines is a subset of a p.m. The concept of n-extendability gradually evolved from the study of elementary bipartite graphs (which are 1-extendable) and then of arbitrary 1-extendable (or 'matching-covered') graphs. The study of n-extendability for arbitrary n was begun by the author (1980). This paper is concerned with matchings in planar graphs. When we speak of an imbedding of planar graph G in the plane, we mean a topological imbedding in the usual sense and would remind the reader that such an imbedding is necessarily 2-cell. If we wish to refer to a planar graph G together with an imbedding of G in the plane, we shall speak of the plane graph G. The main result of this paper is to show that no planar graph is 3-extendable.