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${session.getAttribute("locale")}5Constructions of large graphs on surfaces
https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:18196
Sat 24 Mar 2018 08:04:50 AEDT]]>On large bipartite graphs of diameter 3
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]]>