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Linear Balanceable and Subcubic Balanceable Graphs

Authors


  • Contract grant sponsor: Serbian Ministry of Education and Science; Contract grant number: 174033 (M. R.); Contract grant sponsor: EPSRC; Contract grant number: EP/H021426/1; Contract grant sponsor: Serbian Ministry of Education and Science; Contract grant numbers: 174033; III44006 (K. V.).

  • *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.

Abstract

In Math Program 55(1992), 129–168, 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 ( Discrete Math 310(2010), 3228–3235).

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