Linear Balanceable and Subcubic Balanceable Graphs

In [Structural properties and decomposition of linear balanced matrices, Mathematical Programming, 55:129–168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove that cubic balanced graphs contain a pair of twins, a result that was conjectured by Morris, Spiga and Webb in [Balanced Cayley graphs and balanced planar graphs, Discrete Mathematics, 310:3228–3235, 2010]. ∗Université Paris 7, LIAFA, Case 7014, 75205 Paris Cedex 13, France. E-mail: [email protected]. †Faculty of Computer Science (RAF), Union University, Knez Mihailova 6/VI, 11000 Belgrade, Serbia. E-mail: [email protected]. Partially supported by Serbian Ministry of Education and Science project 174033. ‡CNRS, LIP, ENS Lyon, INRIA, Université de Lyon, 15 parvis René Descartes BP 7000 69342 Lyon cedex 07 France. E-mail: [email protected]. §ENS Lyon, LIP, CNRS, INRIA, Université de Lyon, 15 parvis René Descartes BP 7000 69342 Lyon cedex 07 France. E-mail: [email protected]. ¶School of Computing, University of Leeds, Leeds LS2 9JT, UK and Faculty of Computer Science (RAF), Union University, Knez Mihailova 6/VI, 11000 Belgrade, Serbia. E-mail: [email protected]. Partially supported by EPSRC grant EP/H021426/1 and Serbian Ministry of Education and Science projects 174033 and III44006. The first four authors are partially supported by Agence Nationale de la Recherche under reference anr 10 jcjc 0204 01. All five authors are partially supported by PHC Pavle Savić grant 2010-2011, jointly awarded by EGIDE, an agency of the French Ministère des Affaires étrangères et européennes, and Serbian Ministry of Education and Science.