https://ogma.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:9198 0, that is, (Δ,D,−ε)-graphs. The parameter ε is called the defect. Graphs of defect 1 exist only for Δ = 2. When ε > 1, (Δ,D,−ε)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [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,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 2. Such a catalogue is only the second census of (Δ,D,−2)-graphs known at present, the first being the one of (3,D,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 2 [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 (mod D). 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).]]> Wed 11 Apr 2018 12:46:35 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19509 Sat 24 Mar 2018 08:02:18 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:22080 0, that is, bipartite (Δ,D,−ϵ)-graphs. The parameter ϵ 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>2 bipartite (Δ,D,−ϵ)-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)-graphs with D≥2 and 0≤ϵ≤4; the complete catalogue of bipartite (Δ,D,−ϵ)-graphs with Δ≥2, 5≤D≤187 (D≠6) and 0≤ϵ≤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 [formula could not be replicated].]]> Sat 24 Mar 2018 07:15:16 AEDT ]]>