On the existence of radial Moore graphs for every radius and every degree

Small radial Moore graphs of radius 3

A t-regular graph of radius s is radial Moore if it has diameter at most s + 1 and 1 + t + t(t−1) + . . . + t(t−1)s−1 vertices. We construct radial Moore graphs of radius 3 and degrees t = 3, 5, 7, 9, 10, . . . , 30 with at least t + 1 central vertices and at most t + 2 orbits under the automorphism group.

Radially Moore graphs of radius three and large odd degree

Extremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives some constructions of these extremal graphs for radius three and small degrees. As far as we kn...

Graphs Containing Every 2-Factor

For a graph G, let σ2(G) = min{d(u) + d(v) : uv / ∈ E(G)}. We prove that every n-vertex graph G with σ2(G) ≥ 4n/3−1 contains each 2-regular n-vertex graph. This extends a theorem due to Aigner and Brandt and to Alon and Fisher.

Say every word on every slide

Every Property of Outerplanar Graphs is Testable

A D-disc around a vertex v of a graph G = (V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most n edges, by a set of D-discs around the vertices, for D = D( ) that is independent of the size of the graph. Such a result was already known for ...