A maniplex of rank $n$ is a connected, $n$-valent, edge-coloured graph that generalises abstract polytopes and maps. If the automorphism group of a maniplex $\mathcal{M}$ partitions the vertex-set of $\mathcal{M}$ into $k$ distinct orbits, we say that $\mathcal{M}$ is a $k$-orbit $n$-maniplex. The symmetry type graph of $\mathcal{M}$ is the quotient pregraph obtained by contracting every orbit into a single vertex. Symmetry type graphs of maniplexes satisfy a series of very specific properties. The question arises whether any pregraph of order $k$ satisfying these properties is the symmetry type graph of some $k$-orbit maniplex. We answer the question when $k = 2$.
COBISS.SI-ID: 18761305
In this paper, we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Let $\Gamma$ be a finite graph and let $\operatorname{Aut}(\Gamma )$ be the automorphism group of $\Gamma$. It is well known that we can always find a finite graph $\tilde{\Gamma}$ and a regular covering projection $\wp : \tilde{\Gamma} \to \Gamma$ such that $\operatorname{Aut}(\Gamma )$ lifts along $\wp$. However, for constructing peculiar examples and in applications it is often important, given a subgroup $G$ of $\operatorname{Aut}(\Gamma )$, to find $\wp$ along which $G$ lifts but no further automorphism of $\Gamma$ does, or even that $\operatorname{Aut}(\tilde{\Gamma})$ is the lift of $G$. In this paper, we address these problems.
COBISS.SI-ID: 18761049
We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic $(n_3)$ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the Möbius-Kantor configuration are splittable.
COBISS.SI-ID: 18699097