/manager/Index ${session.getAttribute("locale")} 5 Constructions of large graphs on surfaces /manager/Repository/uon:18196 Sat 24 Mar 2018 08:04:50 AEDT ]]> 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 ]]>