scholarly journals A Note on Sparse Random Graphs and Cover Graphs

10.37236/1497 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tom Bohman ◽  
Alan Frieze ◽  
Miklós Ruszinkó ◽  
Lubos Thoma
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
The Other ◽  
Other Hand ◽  
Cover Graph

It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph $G_{n,p}$ with $p\le \kappa \log n/n$ for any constant $\kappa < 2/3$. On the other hand, this is not true for somewhat higher densities: If $p\ge \lambda (\log n)^3 / (n\log\log n)$ with $\lambda > 1/8$ then with high probability we need to delete more edges than one from every triangle. Our result has a natural algorithmic interpretation.

scholarly journals Between 2- and 3-Colorability

10.37236/4673 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Positive Integer ◽  
Random Graphs ◽  
The Other ◽  
Chromatic Numbers ◽  
Other Hand ◽  
Odd Cycles

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.  These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.

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 When Do Envy-Free Allocations Exist?

2019 ◽  
Vol 33 ◽  
pp. 2109-2116 ◽  
Author(s):  
Pasin Manurangsi ◽  
Warut Suksompong
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Fair Division ◽  
The Other ◽  
Other Hand ◽  
Additive Utilities ◽  
The One

We consider a fair division setting in which m indivisible items are to be allocated among n agents, where the agents have additive utilities and the agents’ utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m = Ω (n log n) but not when m = n + o (n), and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition— instead, the transition is governed by the divisibility relation between m and n. On the one hand, if m is divisible by n, an envy-free allocation exists with high probability as long as m ≥ 2n. On the other hand, if m is not “almost” divisible by , an envy-free allocation is unlikely to exist even when m = Θ(n log n)/log log n).

scholarly journals Packing Tree Factors in Random and Pseudo-Random Graphs

10.37236/3285 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Deepak Bal ◽  
Alan Frieze ◽  
Michael Krivelevich ◽  
Po-Shen Loh
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Edge Disjoint ◽  
Edge Probability ◽  
Vertex Disjoint ◽  
Fixed Tree

For a fixed graph $H$ with $t$ vertices, an $H$-factor of a graph $G$ with $n$ vertices, where $t$ divides $n$, is a collection of vertex disjoint (not necessarily induced) copies of $H$ in $G$ covering all vertices of $G$. We prove that for a fixed tree $T$ on $t$ vertices and $\epsilon>0$, the random graph $G_{n,p}$, with $n$ a multiple of $t$, with high probability contains a family of edge-disjoint $T$-factors covering all but an $\epsilon$-fraction of its edges, as long as $\epsilon^4 n p \gg \log^2 n$. Assuming stronger divisibility conditions, the edge probability can be taken down to $p>\frac{C\log n}{n}$. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

scholarly journals Blocks in Constrained Random Graphs with Fixed Average Degree

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Average Degree ◽  
Additional Restriction ◽  
Structural Constraints ◽  
Analytic Framework ◽  
Discrete Algorithms ◽  
International Audience ◽  
Sharp Concentration

International audience This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.

scholarly journals On the Quantum Chromatic Number of a Graph

10.37236/999 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Ashley Montanaro ◽  
Michael W. Newman ◽  
Simone Severini ◽  
Andreas Winter
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
First Principles ◽  
Clique Number ◽  
The Other ◽  
Quantum Model ◽  
Other Hand ◽  
Orthogonal Representations

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.

Some Reflections on the First Muslim Visitor to Japan

2018 ◽  
Vol 35 (3) ◽  
Author(s):  
James Harry Morris
Keyword(s):  
High Probability ◽  
Research Note ◽  
The Other ◽  
Primary Sources ◽  
Final Part ◽  
Other Hand ◽  
New Hypothesis

This research note explores the biography and visit of the first Muslim visitor to Japan. This figure known as Sādōulǔdīng 撒都魯丁 visited Japan as part of a Mongol envoy in 1275CE, which ultimately ended in his execution in the country. Very little research has been conducted on Sādōulǔdīng, and therefore this note begins by evaluating relevant primary sources in order to assess what information can be garnered about this figure and his visit. Hitherto most scholars have suggested that Sādōulǔdīng was either a Uyghur or an Arab. This note, on the other hand, offers a new hypothesis; namely that there is a high probability that Sādōulǔdīng was of Persian ancestry. In the final part of the note, some thoughts on the significance of his visit to Japan both historically and contemporarily are presented.

scholarly journals Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3338-3389
Author(s):  
Ankit Chauhan ◽  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Maximum Independent Set ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

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 The Spectral Gap of Random Graphs with Given Expected Degrees

10.37236/227 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Amin Coja-Oghlan ◽  
André Lanka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
High Probability ◽  
Spectral Gap ◽  
Sparse Graphs ◽  
Laplacian Eigenvalues

We investigate the Laplacian eigenvalues of a random graph $G(n,\vec d)$ with a given expected degree distribution $\vec d$. The main result is that w.h.p. $G(n,\vec d)$ has a large subgraph core$(G(n,\vec d))$ such that the spectral gap of the normalized Laplacian of core$(G(n,\vec d))$ is $\geq1-c_0\bar d_{\min}^{-1/2}$ with high probability; here $c_0>0$ is a constant, and $\bar d_{\min}$ signifies the minimum expected degree. The result in particular applies to sparse graphs with $\bar d_{\min}=O(1)$ as $n\rightarrow\infty$. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].

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