The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.
COBISS.SI-ID: 1540989892
In this aricle the classification of the integral automorphisms of AG(,) for ?3 is completed.
COBISS.SI-ID: 1538527940
We revisit the configuration of Danzer $DCD(4)$, a great inspiration for our work. This configuration of type $(35_4)$ falls into an infinite series of geometric point-line configurations $DCD(n)$. Each $DCD(n)$ is characterized combinatorially by having the Kronecker cover over the Odd graph $O_n$ as its Levi graph. Danzer's configuration is deeply rooted in Pascal's Hexagrammum Mysticum. Although the combinatorial configuration is highly symmetric, we conjecture that there are no geometric point-line realizations with 7- or 5-fold rotational symmetry; on the other hand, we found a point-circle realization having the symmetry group $D_7$, the dihedral group of order 14.
COBISS.SI-ID: 17492569
We consider a new type of regularity we call edge-girth-regularity. An edge-girth-regular $(v, k, g, \lambda)$-graph $\varGamma$ is a $k$-regular graph of order $v$ and girth $g$ in which every edge is contained in $\lambda$ distinct $g$-cycles. This concept is a generalization of the well-known concept of $(v, k, \lambda)$-edge-regular graphs (that count the number of triangles) and appears in several related problems such as Moore graphs and cage and degree/diameter problems. All edge- and arc-transitive graphs are edge-girth-regular as well. We derive a number of basic properties of edge-girth-regular graphs, systematically consider cubic and tetravalent graphs from this class, and introduce several constructions that produce infinite families of edge-girth-regular graphs. We also exhibit several surprising connections to regular embeddings of graphs in orientable surfaces.
COBISS.SI-ID: 1540315332