/manager/Index ${session.getAttribute("locale")} 5 On large bipartite graphs of diameter 3 /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 ]]> On bipartite graphs of defect at most 4 /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 ]]>