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doi:10.1016/S0012-365X(98)00283-0 
Copyright © 1999 Published by Elsevier Science B.V.
Balanced cycles and holes in bipartite graphs*1
Received 31 March 1997;
revised 1 July 1998;
accepted 22 July 1998. ;
Available online 15 August 2000.
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Abstract
Bruce Reed asks the following question:
Can we determine whether a bipartite graph contains a chordless cycle whose length is a multiple of 4? We show that the two following more general questions are equivalent and we provide an answer. Given a bipartite graph G where each edge is assigned a weight +1 or −1,
- • • determine whether G contains a cycle whose weight is a multiple of 4,
- • • determine whether G contains a chordless cycle whose weight is a multiple of 4.
Given a bipartite graph, we can also decide whether it is possible to assign weights +1 or −1 to its edges so that the above two properties hold.
Author Keywords: Balanced; Hole; Signed graph; Series-parallel graph; 3-path configuration
