An existence result on two-orbit maniplexes Pellicer, Daniel; Potočnik, Primož; Toledo, Micael
Journal of combinatorial theory. Series A,
August 2019, 2019-08-00, Volume:
166
Journal Article
Peer reviewed
Open access
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 M partitions the vertex-set of M into k ...distinct orbits, we say that M is a k-orbit n-maniplex. The symmetry type graph of 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.
It turns out that there exist numerous useful classes of cubic graphs. Some are needed in connection with maps, hypermaps, configurations, polytopes, or covering graphs. In this paper, we briefly ...explore these connections and give motivation why some classes of cubic graphs should be generated. Then we describe the algorithms we used to generate these classes. The results are presented in various tables.
Medial Symmetry Type Graphs Hubard, Isabel; Del Río Francos, María; Orbanić, Alen ...
The Electronic journal of combinatorics,
08/2013, Volume:
20, Issue:
3
Journal Article
Peer reviewed
A $k$-orbit map is a map with its automorphism group partitioning the set of flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbanić, Pellicer and Weiss, for $k \leq 4$. In this paper ...we use symmetry type graphs to extend such study and classify all the types of $5$-orbit maps, as well as all self-dual, properly and improperly, symmetry type of $k$-orbit maps with $k\leq 7$. Moreover, we determine, for small values of $k$, all types of $k$-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest.
Flag graphs have been used in the past for describing maps on closed surfaces. In this paper we use them for the first time in mathematical chemistry for describing benzenoids and some other similar ...structures. Examples include catacondensed and pericondensed benzenoids. Several theorems are included. Symmetries of benzenoid systems, flag graphs, and symmetry type graphs are briefly discussed.
A recently developed theory of flag-graphs and $k$-orbit maps
classifies maps according to their symmetry-type graphs. We
propose a similar classification for polyhedra showing that
Platonic and ...Archimedean solids with the same vertex pattern have
isomorphic symmetry-type graphs and introducing some tools for the
determination of symmetry-type graphs of any polyhedron.
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