1,736 research outputs found
The complex of pant decompositions of a surface
We exhibit a set of edges (moves) and 2-cells (relations) making the complex
of pant decompositions on a surface a simply connected complex. Our
construction, unlike the previous ones, keeps the arguments concerning the
structural transformations independent from those deriving from the action of
the mapping class group. The moves and the relations turn out to be supported
in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of
boundary components), illustrating in this way the so called Grothendieck
principle.Comment: Minor changes in the introductio
Three-coloring triangle-free graphs on surfaces III. Graphs of girth five
We show that the size of a 4-critical graph of girth at least five is bounded
by a linear function of its genus. This strengthens the previous bound on the
size of such graphs given by Thomassen. It also serves as the basic case for
the description of the structure of 4-critical triangle-free graphs embedded in
a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark
On \pi-surfaces of four-dimensional parallelohedra
We show that every four-dimensional parallelohedron P satisfies a recently
found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi
conjecture being true for P. Namely we show that for every four-dimensional
parallelohedron P the group of rational first homologies of its \pi-surface is
generated by half-belt cycles.Comment: 16 pages, 7 figure
- …