Merge branch 'ymogge' into 'master'

new topics from discrete mathematics:

See merge request fabiangabel/research-topics-mat-tuhh!17

.gitlab-ci.yml

@@ -49,6 +49,11 @@ pandoc:
     - pandoc --standalone build/Control_in_Banach_spaces.md -o build/Control_in_Banach_spaces.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
     - pandoc --standalone build/navier-stokes.md -o build/navier-stokes.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
     - pandoc --standalone build/Fractional_Powers_of_Linear_Operators.md -o build/Fractional_Powers_of_Linear_Operators.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
+    - pandoc --standalone build/Connector-Breaker_games_on_random_boards.md -o build/Connector-Breaker_games_on_random_boards.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
+    - pandoc --standalone build/Fast_Strategies_in_Waiter-Client_Games.md -o build/Fast_Strategies_in_Waiter-Client_Games.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
+    - pandoc --standalone build/Maker-Breaker_games_on_randomly_perturbed_graphs.md -o build/Maker-Breaker_games_on_randomly_perturbed_graphs.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
+    - pandoc --standalone build/Random_perturbation_of_sparse_graphs.md -o build/Random_perturbation_of_sparse_graphs.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
+    - pandoc --standalone build/rrr-cross_ttt-intersecting_families_via_necessary_intersection_points.md -o build/rrr-cross_ttt-intersecting_families_via_necessary_intersection_points.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
     - pandoc --standalone build/Zemke_Hessenberg.md -o build/Zemke_Hessenberg.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
     - pandoc --standalone build/PinT-PDEopt.md -o build/PinT-PDEopt.html  --mathjax -c css/base.css -c css/extra.css -c "css/print.css" -c "css/superfish.css" -c "css/superfish-vertical.css" -c "css/base_mode.css"
     # build index

topics/Connector-Breaker_games_on_random_boards.md

+# Connector-Breaker games on random boards
+
+### Working Groups: dm
+
+### Collaborators (MAT): dclemens,  ymogge
+
+### Collaborators (External): Laurin Kirsch
+
+## Description
+
+By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $G\sim G_{n,p}$ is well studied. Recently, London and Pluhár suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. By their results it follows that for this connected version of the game, the threshold bias  on $K_n$ and the threshold probability on $G\sim G_{n,p}$ for winning the game drastically differ from the corresponding values for the usual Maker-Breaker version, assuming Maker's bias to be $1$. However, they observed that the threshold biases of both versions played on $K_n$ are still of the same order if instead Maker is allowed to claim two edges in every round. Naturally, this made London and Pluhár ask whether a similar phenomenon can be observed when a $(2:2)$ game is played on $G_{n,p}$. We prove that this is not the case, and determine the threshold probability for winning this game to be of size $n^{-2/3+o(1)}$.
+
+
+## References
+
+[1] [D. Clemens, L. Kirsch, and Y. Mogge. Connector-Breaker games on random boards, arXiv:1911.01724 (2019), accepted for publication in Electron. J. Combin.](https://arxiv.org/abs/1911.01724)
\ No newline at end of file

topics/Fast_Strategies_in_Waiter-Client_Games.md

+# Fast Strategies in Waiter-Client Games
+
+### Working Groups: dm
+
+### Collaborators (MAT): dclemens, fhamann, pgupta, ymogge
+
+### Collaborators (External): Alexander Haupt, [Mirjana Mikalački](https://people.dmi.uns.ac.rs/~mima/)
+
+## Description
+
+Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.
+
+
+## References
+
+[1] [D. Clemens, P. Gupta, F. Hamann, A. Haupt, M. Mikalački, and Y. Mogge. Fast Strategies in Waiter-Client Games, The Electronic Journal of Combinatorics 27(3) (2020), P3.57](https://doi.org/10.37236/9451 )

topics/Maker-Breaker_games_on_randomly_perturbed_graphs.md

+# Maker-Breaker games on randomly perturbed graphs
+
+### Working Groups: dm
+
+### Collaborators (MAT): dclemens, fhamann, ymogge
+
+### Collaborators (External): [Olaf Parczyk](https://personal.lse.ac.uk/parczyk/)
+
+## Description
+
+Maker-Breaker games are played on a hypergraph $(X,\mathcal{F})$, where $\mathcal{F} \subseteq 2^X$ denotes the family of winning sets. Both players alternately claim a predefined number (called bias) of unclaimed edges from the board $X$, and Maker wins the game if she is able to occupy any winning set $F \in \mathcal{F}$. These games are well studied when played on the complete graph $K_n$ or on a random graph $G_{n,p}$. In this paper we consider Maker-Breaker games played on randomly perturbed graphs instead. These graphs consist of the union of a deterministic graph $G_\alpha$ with minimum degree at least $\alpha n$ and a binomial random graph $G_{n,p}$. 
+Depending on $\alpha$ and Breaker's bias $b$ we determine the order of the threshold probability for winning the Hamiltonicity game and the $k$-connectivity game on $G_{\alpha}\cup G_{n,p}$, and we discuss the $H$-game when $b=1$.
+
+
+## References
+
+[1] [D. Clemens, F. Hamann, Y. Mogge, and O. Parczyk. Maker-Breaker games on randomly perturbed graphs, arXiv:2009.14583 (2020), accepted for publication in SIAM J. Discrete Math.](https://arxiv.org/abs/2009.14583)

topics/Random_perturbation_of_sparse_graphs.md

+# Random perturbation of sparse graphs
+
+### Working Groups: dm
+
+### Collaborators (MAT): ymogge
+
+### Collaborators (External): [Max Hahn-Klimroth](https://www.uni-frankfurt.de/70656959/Max-Hahn-Klimroth?), Giulia S. Maesaka, [Samuel Mohr](https://www.fi.muni.cz/~mohr/), [Olaf Parczyk](https://personal.lse.ac.uk/parczyk/)
+
+## Description
+
+In the model of randomly perturbed graphs we consider the union of a deterministic graph $G_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Posá and Koršunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $G_\alpha \cup \mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha:\mathbb{N} \mapsto (0,1)$ we show that a.a.s. $G_\alpha \cup \mathbb{G}(n,\beta/n)$ is Hamiltonian, where $\beta=−(6+\varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.
+
+
+## References
+
+[1] [M. Hahn-Klimroth, G. Maesaka, Y. Mogge, S. Mohr, and O. Parczyk. Random pertubation of sparse graphs,, arXiv:2004.04672 (2020)](https://arxiv.org/abs/2004.04672)
\ No newline at end of file

topics/rrr-cross_ttt-intersecting_families_via_necessary_intersection_points.md

+# $r$-cross $t$-intersecting families via necessary intersection points
+
+### Working Groups: dm
+
+### Collaborators (MAT): pgupta, ymogge
+
+### Collaborators (External): Simón Piga, [Bjarne Schülke](https://www.math.uni-hamburg.de/home/schuelke/)
+
+## Description
+
+Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathscr{P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of $r$-cross $t$-intersecting families. This also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for families of possibly mixed uniformities $k_1,\ldots,k_r$. 
+
+
+## References
+
+[1] [P. Gupta, Y. Mogge, S. Piga, and B. Schülke. $r$-cross $t$-intersecting families via necessary intersection points, arXiv:2010.11928 (2020)](https://arxiv.org/abs/2010.11928)
\ No newline at end of file