Perfect Matching in General vs. Cubic Graphs: A Note on the Planar and Bipartite Cases

¹ LaMI, Université d'Evry, boulevard des Coquibus, 91025 Evry Cedex, France; (bampis@lami.univ-evry.fr)
² La.R.I.A., Université de Picardie-Jules Verne, 5 rue du Moulin Neuf, 80000 Amiens, France; (aristote@laria.u-picardie.fr)
³ Institute for System Analysis, 9, Prosp. 60 Let Octyabrya, 117312 Moscow, Russia.
⁴ LRI, bâtiment 490, Université Paris-Sud, 91405 Orsay Cedex, France; (yannis@lri.fr)
⁵ Athens University of Economics and Business, Department of Informatics, 76 Patission St., 10434 Athens, Greece; (milis@aueb.gr)

Received: 24 March 1997
Accepted: 21 March 2000

Abstract

It is known that finding a perfect matching in a general graph is AC⁰-equivalent to finding a perfect matching in a 3-regular (i.e. cubic) graph. In this paper we extend this result to both, planar and bipartite cases. In particular we prove that the construction problem for perfect matchings in planar graphs is as difficult as in the case of planar cubic graphs like it is known to be the case for the famous Map Four-Coloring problem. Moreover we prove that the existence and construction problems for perfect matchings in bipartite graphs are as difficult as the existence and construction problems for a weighted perfect matching of O(m) weight in a cubic bipartite graph.