[Facets-of-complexity] Invitation & Link to Monday Lecture & Colloquium - April 26th 2021

[Ita Brunke <i.brunke@inf.fu-berlin.de>]

You are cordially invited to our next Monday Lecture & Colloquium on April 26th at 14:15 h & 16:00 h, online via Zoom.

Invitation link:
https://tu-berlin.zoom.us/j/69716124232?pwd=dzFlcTFHMmFXRTE5QmZLaEV5N0FRUT09
No password required.

Online via:
Zoom - Invitation

Time: Monday, April 26th - 14:15 h

Lecture: Kolja Knauer (Universitat de Barcelona)

Title: On sensitivity in Cayley graphs

Abstract:

Recently, Huang proved the Sensitivity Conjecture, by showing that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$. This is tight by a result of Chung, F\"uredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. In this lecture we study Huang's question on Cayley graphs of groups.

We show that high symmetry alone does not guarantee similar behavior and present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are $d$-regular containing an induced matching on a $\frac{d}{2d-1}$-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.

On the positive side, we consider Cayley graphs of Coxeter groups, where a lower bound similar to Huang's can be shown. A generalization of the construction of Chung, F\"uredi, Graham, and Seymour shows that this bound is tight for products of Coxeter groups of type $\mathbf{A_n}$, $\mathbf{I_n}(2k+1)$, most exceptional cases and not far from optimal in general.
Then, we show that also induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This yields more classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no large subcubes.
Joint with Ignacio Garcia-Marco.

Break

Time: Monday, April 26th - 16:00 h s.t.

Colloquium: Matthieu Rosenfeld (LIRMM)

Title: Bounding the number of sets defined by a given MSO formula on trees

Abstract:

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets, or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order n is at most 95^(n/13) and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second-order formula.
I will first illustrate the general technique with a really simple concrete example ( Dominating independent sets). Then I will explain how to generalize this into a more general technique. I will end my talk by mentioning a few of the results obtained with this technique.