14 research outputs found
Transitive path decompositions of Cartesian products of complete graphs
An -decomposition of a graph is a partition of its edge set into
subgraphs isomorphic to . A transitive decomposition is a special kind of
-decomposition that is highly symmetrical in the sense that the subgraphs
(copies of ) are preserved and transitively permuted by a group of
automorphisms of . This paper concerns transitive -decompositions of
the graph where is a path. When is an odd prime, we
present a construction for a transitive path decomposition where the paths in
the decomposition are arbitrary large.Comment: 14 pages, 4 figure
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Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences
Arithmetic groups arise naturally in many fields such as number theory, algebraic geometry, and analysis. Mapping class groups arise in both low dimensional topology and geometric group theory. They have been studied intensively by different groups of people. The purpose of this workshop is to bring experts and aspiring young mathematicians together to interact and develop further exchanges and new collaboration
The Galois action on Origami curves and a special class of Origamis
Origamis are covers of elliptic curves, ramified over at most one point. As they admit a flat atlas, they induce Teichmüller curves in the corresponding moduli spaces of curves. We compare geometric and arithmetic properties of these objects and study in detail a construction given by M. Möller which associates an Origami to a Belyi morphism. This leads to new examples, such as Galois orbits of Origami curves, and an infinite series of non-characteristic Origamis with Veech group SL(2,Z)
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum