scholarly journals Connector-Breaker Games on Random Boards

10.37236/9381 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Laurin Kirsch ◽  
Yannick Mogge
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Threshold Probability

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

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Some remarks about extreme degrees in a random graph

1986 ◽  
Vol 100 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Probability Distributions ◽  
Supplementary Information ◽  
Random Subgraph

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

scholarly journals Small Subgraphs in the Trace of a Random Walk

10.37236/6169 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Peleg Michaeli
Keyword(s):  
Random Walk ◽  
Random Graph ◽  
Complete Graph ◽  
Combinatorial Properties ◽  
Base Graph

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.

scholarly journals Ramsey Games Against a One-Armed Bandit

2003 ◽  
Vol 12 (5-6) ◽  
pp. 515-545 ◽  
Author(s):  
Ehud Friedgut ◽  
Yoshiharu Kohayakawa ◽  
Vojtěch Rodl ◽  
Andrzej Rucinski ◽  
Prasad Tetali
Keyword(s):  
Random Graph ◽  
Positive Constant ◽  
Complete Graph ◽  
Upper Bound ◽  
High Probability ◽  
Random Sequence ◽  
Extreme Case ◽  
Online Version ◽  
Expected Length ◽  
Person Game

We study the following one-person game against a random graph process: the Player's goal is to 2-colour a random sequence of edges of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.

scholarly journals Diameter of the Stochastic Mean-Field Model of Distance

2017 ◽  
Vol 26 (6) ◽  
pp. 797-825 ◽  
Author(s):  
SHANKAR BHAMIDI ◽  
REMCO VAN DER HOFSTAD
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Field Model ◽  
Mean Field ◽  
Random Variable ◽  
Weak Limit ◽  
Random Structure ◽  
Mean Field Model ◽  
Graph Diameter

We consider the complete graph 𝜅n on n vertices with exponential mean n edge lengths. Writing Cij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that maxi,j∈[n]Cij/logn converges in probability to 3. We extend these results by showing that maxi,j∈[n]Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdős–Rényi random graph in [22].

Cliques in random graphs

1976 ◽  
Vol 80 (3) ◽  
pp. 419-427 ◽  
Author(s):  
B. Bollobas ◽  
P. Erdös
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Random Variables ◽  
Natural Numbers ◽  
Complete Subgraph ◽  
Maximal Size ◽  
Point Set ◽  
Image Position

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.

scholarly journals Client-Waiter Games on Complete and Random Graphs

10.37236/6039 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Oren Dean ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Upper Bounds ◽  
Connected Component ◽  
Lower And Upper Bounds ◽  
Large Connected Component ◽  
Graph Property ◽  
Player Game

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

scholarly journals A Remark on the Tournament Game

10.37236/5142 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Mirjana Mikalački
Keyword(s):  
Random Graph ◽  
Threshold Probability

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, while Maker additionally assigns orientations to the edges that she claims. If by the end of the game Maker claims all the edges of a pre-defined goal tournament, she wins the game. 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. 

Random trees in a graph and trees in a random graph

1986 ◽  
Vol 100 (2) ◽  
pp. 319-330 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Random Trees ◽  
Labelled Trees

This paper treats two related sets of problems in the theory of random graphs. In Sections 2 and 3 we study random spanning subtrees of a complete graph (or, equivalently, random labelled trees). It is shown that the number of common edges of two such random trees asymptotically has a Poisson distribution with expectation 2. Similar results are obtained for the number of edges in the intersection or union of more than two random trees.

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