arXiv subjects

Dennis Clemens

Publications and source records attributed to Dennis Clemens.

35 records · Page 2Linked to original sources

arXiv2020

On minimal Ramsey graphs and Ramsey equivalence in multiple colours

For an integer $q\ge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Nešetřil and Rödl from 1977 we prove that, for $q\ge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences. (1) For $2\le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$. (2) For every $q\ge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection $\{{\cal M}_q(H) : H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where ${\cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$. We also address the question which pairs of graphs satisfy ${\cal M}_q(H_1)={\cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $q\ge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

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arXiv2019

Connector-Breaker games on random boards

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 $n^{-2/3+o(1)}$.

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arXiv2019

Perfectly packing graphs with bounded degeneracy and many leaves

We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

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arXiv2019

A Dirac-type theorem for Berge cycles in random hypergraphs

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the following Dirac-type theorem about Berge cycles in the binomial random $r$-uniform hypergraph $H^{(r)}(n,p)$: for every integer $r \geq 3$, every real $γ>0$ and $p \geq \frac{\ln^{17r} n}{n^{r-1}}$ asymptotically almost surely, every spanning subgraph $H \subseteq H^{(r)}(n,p)$ with minimum vertex degree $δ_1(H) \geq \left(\frac{1}{2^{r-1}} + γ\right) p \binom{n}{r-1}$ contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on $p$ is optimal up to some polylogarithmic factor.

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arXiv2018

Colourings without monochromatic disjoint pairs

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős-Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure. The celebrated Erdős-Ko-Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős-Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces. In this paper we present a unified approach to the Erdős-Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős-Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann.

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arXiv2017

The size-Ramsey number of powers of paths

Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is $q$-Ramsey for $H$, denoted $G\rightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is defined to be $\hat{r}(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}$. Answering a question of Conlon, we prove that, for every fixed $k$, we have $\hat{r}(P_n^k)=O(n)$, where $P_n^k$ is the $k$-th power of the $n$-vertex path $P_n$ (i.e. , the graph with vertex set $V(P_n)$ and all edges $\{u,v\}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.

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arXiv2016

A non-trivial upper bound on the threshold bias of the Oriented-cycle game

In the Oriented-cycle game, introduced by Bollobás and Szabó, two players, called OMaker and OBreaker, alternately direct edges of $K_n$. OMaker directs exactly one edge, whereas OBreaker is allowed to direct between one and $b$ edges. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. Bollobás and Szabó conjectured that for a bias as large as $n-3$ OMaker has a winning strategy if OBreaker must take exactly $b$ edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and Sudakov, that OMaker has a winning strategy for this game whenever $b\leq \frac{n}{2}-2$. In this paper, we show that OBreaker has a winning strategy whenever $b\geq \frac{5n}{6}+2$. Moreover, in case OBreaker is required to direct exactly $b$ edges in each move, we show that OBreaker wins for $b\geq \frac{19n}{20}$, provided that $n$ is large enough. This refutes the conjecture by Bollobás and Szabó.

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arXiv2016

Creating cycles in Walker-Breaker games

We consider biased $(1:b)$ Walker-Breaker games: Walker and Breaker alternately claim edges of the complete graph $K_n$, Walker taking one edge and Breaker claiming $b$ edges in each round, with the constraint that Walker needs to choose her edges according to a walk. As questioned in a paper by Espig, Frieze, Krivelevich and Pegden, we study how long a cycle Walker is able to create and for which biases $b$ Walker has a chance to create a cycle of given constant length.

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arXiv2016

How fast can Maker win in fair biased games?

We study (a:a) Maker-Breaker games played on the edge set of the complete graph on n vertices. In the following four games - perfect matching game, Hamilton cycle game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how Red can win in the strong version of these games.

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arXiv2015

On sets not belonging to algebras and rainbow matchings in graphs

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$ elements whose equivalence classes with respect to $A_i$ are nontrivial, then $A_1, \ldots, A_n$ contain a rainbow matching, i.e. there exist $2n$ distinct elements $x_1,y_1,\ldots,x_n,y_n\in X$ with $x_i\sim_{A_i} y_i$ for each $i\in [n]$. Grinblat asked whether $\mathfrak{v}(n) = 3n-2$ for every $n\geq 4$. The best-known upper bound was $\mathfrak{v}(n) \leq 16n/5 + \mathcal{O}(1)$ due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that $\mathfrak{v}(n) = 3n+o(n)$.

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arXiv2015

A remark on the Tournament game

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined goal tournament. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.

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arXiv2015

An improved bound on the sizes of matchings guaranteeing a rainbow matching

A conjecture by Aharoni and Berger states that every family of $n$ matchings of size $n+1$ in a bipartite multigraph contains a rainbow matching of size $n$. In this paper we prove that matching sizes of $(3/2 + o(1)) n$ suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best known one in case we only aim to find a rainbow matching of size $n-1$. This improves previous results by Aharoni, Charbit and Howard, and Kotlar and Ziv.

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arXiv2015

Minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs

A uniform hypergraph $H$ is called $k$-Ramsey for a hypergraph $F$, if no matter how one colors the edges of $H$ with $k$ colors, there is always a monochromatic copy of $F$. We say that $H$ is minimal $k$-Ramsey for $F$, if $H$ is $k$-Ramsey for $F$ but every proper subhypergraph of $H$ is not. Burr, Erdős and Lovasz studied various parameters of minimal Ramsey graphs. In this paper we initiate the study of minimum degrees and codegrees of minimal Ramsey $3$-uniform hypergraphs. We show that the smallest minimum vertex degree over all minimal $k$-Ramsey $3$-uniform hypergraphs for $K_t^{(3)}$ is exponential in some polynomial in $k$ and $t$. We also study the smallest possible minimum codegrees over minimal $2$-Ramsey $3$-uniform hypergraphs.

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arXiv2014

Keeping Avoider's graph almost acyclic

We consider biased $(1:b)$ Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if $b \geq 200 n \ln n$. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-$k$-colorability game, and the $K_t$-minor game thus addressing a question and improving the results of Hefetz, Krivelevich, Stojaković, and Szabó. Moreover, we give a slight improvement for the lower bound in the non-planarity game.

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arXiv2013

The random graph intuition for the tournament game

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph on n vertices and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2-o(1))log_2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two "clever" players and the game played by two "random" players, is supported by the tournament game: It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation-game version of the tournament game, where Maker wins the game if the final digraph -- containing also the edges directed by Breaker -- possesses a copy of T_k. We prove that in that game Breaker has a winning strategy for k = (4+o(1))log_2 n.

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arXiv2013

Building spanning trees quickly in Maker-Breaker games

For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.

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arXiv2012

Fast strategies in Maker-Breaker games played on random boards

In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.

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