scholarly journals Strong Games Played on Random Graphs

10.37236/5414 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Asaf Ferber ◽  
Pascal Pfister
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Perfect Matching ◽  
Hamilton Cycle ◽  
Target Structure ◽  
Fixed Constant ◽  
Matching Game

In a strong game played on the edge set of a graph $G$ there are two players, Red and Blue, alternating turns in claiming previously unclaimed edges of $G$ (with Red playing first). The winner is the first one to claim all the edges of some target structure (such as a clique $K_k$, a perfect matching, a Hamilton cycle, etc.). In this paper we consider strong games played on the edge set of a random graph $G\sim G(n,p)$ on $n$ vertices. We prove that $G\sim G(n,p)$ is typically such that Red can win the perfect matching game played on $E(G)$, provided that $p\in(0,1)$ is a fixed constant. 

scholarly journals Random-Player Maker-Breaker games

10.37236/5032 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Krivelevich ◽  
Gal Kronenberg
Keyword(s):  
Perfect Matching ◽  
Winning Strategy ◽  
Hamilton Cycle ◽  
The Other ◽  
Natural Question ◽  
Vertex Connectivity ◽  
Asymptotic Values ◽  
Player Game ◽  
Matching Game ◽  
As If

In a $(1:b)$ Maker-Breaker game, one of the central questions is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ (or $m$) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge set of $K_n$). For each of these games we find or estimate the asymptotic values of the bias (either $b$ or $m$) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.

scholarly journals Multicoloured Hamilton cycles in random graphs; an anti-Ramsey threshold

10.37236/1213 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hamilton Cycle ◽  
Hamilton Cycles

Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer to this as a $k$-bounded colouring. We say that a subset of the edges of $G$ is multicoloured if each edge is of a different colour. We say that the colouring is $\cal H$-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ${\cal AR}_k$ = $\{G :$ every $k$-bounded colouring of $G$ is $\cal H$-good$\}$. We establish the threshold for the random graph $G_{n,m}$ to be in ${\cal AR}_k$.

scholarly journals Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

10.37236/8279 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Daniela Kühn ◽  
Deryk Osthus ◽  
Jaehoon Kim
Keyword(s):  
Random Graphs ◽  
Perfect Matching ◽  
Degree Sequence ◽  
Hamilton Cycle ◽  
Vertex Degree ◽  
Degree Sequences ◽  
Hamilton Cycles ◽  
Degree Condition ◽  
Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

scholarly journals Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

10.37236/5025 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Asaf Ferber
Keyword(s):  
Random Graph ◽  
Directed Graph ◽  
Random Graphs ◽  
High Probability ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Edge Probability

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd $n$ their $p$ was $\omega((\log n)/n)$. Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we randomly color the edge set of a random directed graph $D_{n,p}$ with $(1+o(1))n$ colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.

scholarly journals Random Perturbation of Sparse Graphs

10.37236/9510 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Max Hahn-Klimroth ◽  
Giulia Maesaka ◽  
Yannick Mogge ◽  
Samuel Mohr ◽  
Olaf Parczyk
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Random Perturbation ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Fixed Constant ◽  
Bounded Degree Trees ◽  
Random Part

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ 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 Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ 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 $\mathbb{G}(n,p)$.

scholarly journals The Threshold for the Full Perfect Matching Color Profile in a Random Coloring of Random Graphs

10.37236/9066 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Debsoumya Chakraborti ◽  
Mihir Hasabnis
Keyword(s):  
Bipartite Graph ◽  
Random Graph ◽  
Random Graphs ◽  
Perfect Matching ◽  
Color Profile ◽  
Matching Color ◽  
Random Coloring

Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \{c_1,c_2, \ldots, c_q\}$. Let $Q_i$ be the set of all edges colored with $c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\mathrm{mcp}(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q)$ such that there exists a perfect matching $M$ in $G$ with $|Q_i \cap M| = m_i$ for all $i$. Let $\alpha_1, \alpha_2, \ldots, \alpha_q$ be positive constants such that $\sum_{i=1}^q \alpha_i = 1$. Let $G$ be the random bipartite graph $G_{n,n,p}$. Suppose the edges of $G$ are independently colored with color $c_i$ with probability $\alpha_i$. We determine the threshold for the event $\mathrm{mcp}(G) = \{(m_1, \ldots, m_q) \in [0,n]^q : m_1 + \cdots + m_q = n\},$ answering a question posed by Frieze. We further extend our methods to find the threshold for the same event in a randomly colored random graph $G_{n,p}$.  

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

scholarly journals Sharp Threshold Functions for Random Intersection Graphs via a Coupling Method

10.37236/523 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Perfect Matching ◽  
Hamilton Cycle ◽  
Coupling Method ◽  
New Method ◽  
Intersection Graphs ◽  
Sharp Threshold ◽  
Threshold Functions

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for $k$–connectivity, perfect matching containment and Hamilton cycle containment.

scholarly journals Connectivity for Random Graphs from a Weighted Bridge-Addable Class

10.37236/2596 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Additional Assumption ◽  
General Distribution ◽  
Expected Number ◽  
Recent Interest ◽  
Main Conjecture

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components   then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a   class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

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