1 research outputs found
Minimum k-critical bipartite graphs
We study the problem of Minimum -Critical Bipartite Graph of order
- MCBG-: to find a bipartite , with , , and
, which is -critical bipartite, and the tuple , where and denote the maximum degree in
and , respectively, is lexicographically minimum over all such graphs.
is -critical bipartite if deleting at most vertices from creates
that has a complete matching, i.e., a matching of size . We show that,
if is an integer, then a solution of the MCBG- problem
can be found among -regular bipartite graphs of order , with
, and . If , then all -regular bipartite
graphs of order are -critical bipartite. For , it is not the
case. We characterize the values of , , , and that admit an
-regular bipartite graph of order , with , and give a
simple construction that creates such a -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