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Documents  05C10 | enregistrements trouvés : 7

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Post-edited  Coloring graphs on surfaces
Esperet, Louis (Auteur de la Conférence) | CIRM (Editeur )

- Normalized characters of the symmetric groups,
- Kerov polynomials and Kerov positivity conjecture,
- Stanley character polynomials and multirectangular coordinates of Young diagrams,
- Stanley character formula and maps,
- Jack characters
- characterization, partial results.

05E10 ; 05E15 ; 20C30 ; 05A15 ; 05C10

The partially disjoint paths problem asks for paths $P_1, \ldots,P_k$ between given pairs of terminals, while certain pairs of paths $P_i$,$P_j$ are required to be disjoint. With the help of combinatorial group theory, we show that, for fixed $k$, this problem can be solved in polynomial time for planar directed graphs. We also discuss related problems. No specific foreknowledge is required.

05C10 ; 05C20 ; 05C25 ; 05C38 ; 68Q25 ; 90C27

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.
Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $O(g^{3/2}n^{1/2})$ for any triangulated combinatorial surface of genus g with n triangles, and describe an $O(gn)$-time algorithm to compute such a decomposition.
Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
systolic geometry - computational topology - topological graph theory - graphs on surfaces - triangulations - random graphs
How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed ...

05C10 ; 68U05 ; 53C23 ; 57M15 ; 68R10

Multi angle  Embedding extension problems
Mohar, Bojan (Auteur de la Conférence) | CIRM (Editeur )

Multi angle  Vertex degrees in planar maps
Drmota, Michael (Auteur de la Conférence) | CIRM (Editeur )

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even inf...

05A19 ; 05A16 ; 05C10 ; 05C30

Multi angle  Random cubic planar graphs revisited
Rué, Juanjo (Auteur de la Conférence) | CIRM (Editeur )

We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show that the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance, while formerly it was only known that it is linear with high probability.
This is based on a joint work with Marc Noy (UPC) and Clément Requilé (FU Berlin - BMS).
We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show ...

05C80 ; 05C10 ; 05A16

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