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${session.getAttribute("locale")}5On large bipartite graphs of diameter 3
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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
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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]]>