On bipartite graphs of diameter 3 and defect 2

  • Authors:

  • Charles Delorme;Leif K. Jørgensen;Mirka Miller;Guillermo Pineda-Villavicencio

  • Affiliations:

  • Lab. de Recherche en Informatique, University de Paris-Sud, 91405 Orsay, France;Department of Mathematics and Computer Science, Aalborg University, F. Bajers Vej 7, DK-9220 Aalborg Ø, Denmark;School of Electrical Engineering and Computer Science, The University of Newcastle, Australia, Newcastle, Australia and Department of Mathematics, University of West Bohemia, Pilsen, Czech Republi ...;School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Australia and Department of Computer Science, University of Oriente, Santiago de Cuba, Cuba

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2009

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Abstract

We consider bipartite graphs of degree Δ≥2, diameter D=3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (Δ, 3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (Δ, 3, -2) -graphs. The most general of these conditions is that either Δ or Δ-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when Δ=6 and Δ=9, we prove the non-existence of the corresponding bipartite (Δ, 3, -2)-graphs, thus establishing that there are no bipartite (Δ, 3, -2)-graphs, for 5≤Δ≤10. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 271–288, 2009