http://researchonline.federation.edu.au/vital/access/manager/Index ${session.getAttribute("locale")} 5 http://researchonline.federation.edu.au/vital/access/manager/Repository/vital:6894 Wed 07 Apr 2021 13:46:26 AEST ]]> http://researchonline.federation.edu.au/vital/access/manager/Repository/vital:4958 Wed 07 Apr 2021 13:44:33 AEST ]]> http://researchonline.federation.edu.au/vital/access/manager/Repository/vital:4267 0, that is, bipartite (Δ, D, - ε{lunate})-graphs. The parameter ε{lunate} is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if Δ ≥ 3 and D ≥ 3, they may only exist for D = 3. However, when ε{lunate} > 2 bipartite (Δ, D, - ε{lunate})-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (Δ, D, - 4)-graphs; the complete catalogue of bipartite (3, D, - ε{lunate})-graphs with D ≥ 2 and 0 ≤ ε{lunate} ≤ 4; the complete catalogue of bipartite (Δ, D, - ε{lunate})-graphs with Δ ≥ 2, 5 ≤ D ≤ 187 (D ≠ 6) and 0 ≤ ε{lunate} ≤ 4; a proof of the non-existence of all bipartite (Δ, D, - 4)-graphs with Δ ≥ 3 and odd D ≥ 5. Finally, we conjecture that there are no bipartite graphs of defect 4 for Δ ≥ 3 and D ≥ 5, and comment on some implications of our results for the upper bounds of Nb (Δ, D). © 2011 Elsevier B.V. All rights reserved.]]> Wed 07 Apr 2021 13:43:42 AEST ]]> http://researchonline.federation.edu.au/vital/access/manager/Repository/vital:4236 0, that is, (Δ,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Δ=2. When >1, (Δ,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-Purón and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Δ,D,-2)-graph with Δ<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Δ,D,-2)-graphs with even Δ<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0≤≤2. Such a catalogue is only the second census of (Δ,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0≤≤2 Jørgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Δ,D,-2)-graphs with odd Δ<5 and D<4, and the non-existence of (Δ,D,-2)-graphs with odd Δ<5 and D<5 such that Δ≡0,2(modD). Finally, we conjecture that there are no (Δ,D,-2)-graphs with Δ<4 and D<4, and comment on some implications of our results for the upper bounds of N(Δ,D). © 2011 Elsevier B.V. All rights reserved.]]> Wed 07 Apr 2021 13:43:40 AEST ]]>