  In [{Structural properties and decomposition of linear balanced matrices},
{\it 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,
{\it Discrete Mathematics}, 310:3228--3235, 2010].
