Linear balanceable and subcubic balanceable graphs

TL;DR

This paper proves a conjecture about the existence of non-unique cycle chords in certain balanced bipartite graphs, extending results to linear and subcubic balanceable graphs and confirming a conjecture about twins in cubic balanced graphs.

Contribution

It establishes the conjecture for linear balanced bipartite graphs without 4-cycles and for graphs with maximum degree 3, also proving the existence of twins in cubic balanced graphs.

Findings

Proved the conjecture for linear balanced bipartite graphs without 4-cycles.

Extended results to subcubic balanceable graphs.

Confirmed the existence of twins in cubic balanced graphs.

Abstract

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].