Minimum [formula omitted]-critical bipartite graphs

Abstract

We study the problem of Minimum k-Critical Bipartite Graph of order (n,m) - MkCBG-(n,m): to find a bipartite G=(U,V;E), with |U|=n, |V|=m, and n>m>1, which is k-critical bipartite, and the tuple (|E|,ΔU,ΔV), where ΔU and ΔV denote the maximum degree in U and V, respectively, is lexicographically minimum over all such graphs. G is k-critical bipartite if deleting at most k=n−m vertices from U creates G′ that has a complete matching, i.e., a matching of size m. We show that, if m(n−m+1)∕n is an integer, then a solution of the MkCBG-(n,m) problem can be found efficiently among (a,b)-regular bipartite graphs of order (n,m), with a=m(n−m+1)∕n, and b=n−m+1. If a=m−1, then all (a,b)-regular bipartite graphs of order (n,m) are k-critical bipartite. For a<m−1, it is not the case. We characterize the values of n, m, a, and b that admit an (a,b)-regular bipartite graph of order (n,m), with b=n−m+1, and give a simple construction that creates such a k-critical bipartite graph whenever possible. Our techniques are based on Hall’s marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.