scholarly journals Triangle resilience of the square of a Hamilton cycle in random graphs

2022 ◽  
Vol 152 ◽  
pp. 171-220
Author(s):  
Manuela Fischer ◽  
Nemanja Škorić ◽  
Angelika Steger ◽  
Miloš Trujić
Keyword(s):  
Random Graphs ◽  
Hamilton Cycle

scholarly journals On two Hamilton cycle problems in random graphs

2008 ◽  
Vol 166 (1) ◽  
pp. 221-234 ◽  
Author(s):  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Random Graphs ◽  
Hamilton Cycle

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 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.

Applications of random graphs

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Social Networks ◽  
Random Graphs ◽  
Interaction Network ◽  
Power Grids ◽  
Protein Protein Interaction ◽  
Topological Features ◽  
First Case ◽  
The Stability ◽  
Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

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.

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