https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 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:28645 Tue 18 Jul 2017 11:23:58 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7792 Sat 24 Mar 2018 08:39:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7828 Sat 24 Mar 2018 08:37:39 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7826 Sat 24 Mar 2018 08:37:39 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7820 Sat 24 Mar 2018 08:37:36 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:8137 Sat 24 Mar 2018 08:36:09 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:18196 Sat 24 Mar 2018 08:04:50 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19421 degree/diameter problem, namely, given natural numbers d≥2 and D≥2, find the maximum number _{Nᵇ(d, D)} of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound _Mᵇ(d,D) represents a general upper bound for _Nᵇ(d,D). Bipartite graphs of order _Mᵇ(d,D) are very rare, and determining _Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,−4)-graphs (that is, bipartite graphs of order _Mb(d,D)−4) was carried out. Here we first present some structural properties of bipartite (d,3,−4)-graphs, and later prove that there are no bipartite (7,3,−4)-graphs. This result implies that the known bipartite (7,3,−6)-graph is optimal, and therefore _Nᵇ(7,3)=80. We dub this graph the Hafner–Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,−4)-graph, and the non-existence of bipartite (6,3,−4)-graphs. In addition, we discover at least one new largest known bipartite–and also vertex-transitive–graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for _Nᵇ(11,3).]]> Sat 24 Mar 2018 07:51:59 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23514 Sat 24 Mar 2018 07:17:17 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 ]]>