On Minimum Sets of 1-Factors Covering a Complete Multipartite Graph David Cariolaro 1 and Hung-Lin Fu 2 1 INSTITUTE OF MATHEMATICS ACADEMIA SINICA, NANKANG, TAIPEI 11529, TAIWAN E-mail: cariolaro@math.sinica.edu.tw; davidcariolaro@hotmail.com 2 DEPARTMENT OF APPLIED MATHEMATICS NATIONAL CHIAO TUNG UNIVERSITY HSIN CHU 30050 TAIWAN E-mail: hlfu@math.nctu.edu.tw Received April 20, 2006; Revised January 21, 2008 Published online 15 April 2008 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20303 Abstract: We determine necessary and sufficient conditions for a com- plete multipartite graph to admit a set of 1-factors whose union is the whole graph and, when these conditions are satisfied, we determine the minimum size of such a set. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 239–250, 2008 Keywords: 1–factor; 1–factor cover; excessive factorization; excessive index; complete multipartite graph 1. INTRODUCTION All graphs considered will be finite, simple and undirected, unless stated otherwise. We denote by V (G) and E(G), respectively, the vertex and edge set of a graph G. Journal of Graph Theory © 2008 Wiley Periodicals, Inc. 239