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On Random Graph Homomorphisms into Z

2000
*
Journal of combinatorial theory. Series B (Print)
*

Given a bipartite connected nite

doi:10.1006/jctb.1999.1931
fatcat:5prkyt4ktnhohpr3m6gjsol5y4
*graph*G = (V; E) and a vertex v 0 2 V , we consider uniform probability measure*on*the set of*graph**homomorphisms*f : V !*Z*satisfying f (v 0 ) = 0. ... by the distance to 0 of a simple*random*walk*on**Z*having run for d(u; v) steps. ... Thanks to Yuval Peres, Je Steif and Oded Schramm for useful discussions, and to Johan Jonasson for helpful comments*on*the manuscript. ...##
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Random cubic graphs are not homomorphic to the cycle of size 7

2005
*
Journal of combinatorial theory. Series B (Print)
*

We prove that a

doi:10.1016/j.jctb.2004.11.004
fatcat:66bmajzgijgabfb4qfhb37iraq
*random*cubic*graph*almost surely is not*homomorphic*to a cycle of size 7. ... This implies that there exist cubic*graphs*of arbitrarily high girth with no*homomorphisms*to the cycle of size 7. ... Acknowledgments The author thanks Michael Molloy for his valuable discussions leading towards the result of this paper and his valuable comments*on*the draft version, and Xuding Zhu for introducing the ...##
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Random cubic graphs are not homomorphic to the cycle of size 7
[article]

2006
*
arXiv
*
pre-print

We prove that a

arXiv:math/0701013v1
fatcat:bpxwf3cwwfapjb4mcdhzdt44s4
*random*cubic*graph*almost surely is not*homomorphic*to a cycle of size 7. ... This implies that there exist cubic*graphs*of arbitrarily high girth with no*homomorphisms*to the cycle of size 7. ... Acknowledgment The author wishes to thank Michael Molloy for his valuable discussions leading towards the result of this paper and his valuable comments*on*the draft version, and Xuding Zhu for introducing ...##
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Removal lemmas and approximate homomorphisms
[article]

2022
*
arXiv
*
pre-print

*One*such variant, which we call the triangle-free lemma, states that for each ϵ>0 there exists M such that every triangle-free

*graph*G has an ϵ-approximate

*homomorphism*to a triangle-free

*graph*F

*on*at ... most M vertices (here an ϵ-approximate

*homomorphism*is a map V(G) → V(F) where all but at most ϵ |V(G)|^2 edges of G are mapped to edges of F). ... We thank Yuval Wigderson and the anonymous reviewer for careful readings and comments

*on*the manuscript. ...

##
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A cluster expansion approach to exponential random graph models

2012
*
Journal of Statistical Mechanics: Theory and Experiment
*

The exponential family of

doi:10.1088/1742-5468/2012/05/p05004
fatcat:r74lpvsjzbdwbpenfq53knk4bu
*random**graphs*is among the most widely-studied network models. ... We show that any exponential*random**graph*model may alternatively be viewed as a lattice gas model with a finite Banach space norm. ... Step 2: We check whether these vertex-maps are valid*homomorphisms*(i.e., edge-preserving).*Graphs*in the same exponential*random**graph*family correspond to equilibrium ensembles. ...##
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On the derandomization of the graph test for homomorphism over groups

2011
*
Theoretical Computer Science
*

For abelian groups G =

doi:10.1016/j.tcs.2010.12.046
fatcat:zbw6oizap5cetlcbvc5ys7v3za
*Z*m p , Γ =*Z*p and function f : G → Γ , by using a λ-biased set S of size poly(log |G|), we show that,*on*any given bipartite*graph*H = (V 1 , V 2 ; E), there exists a*graph*test ... For general groups G, Γ and function f : G → Γ , we introduce k*random*walks of some length, ℓ say,*on*expander*graphs*to design a probabilistic*homomorphism*test, which could be thought as a*graph*test ... For general groups G, Γ and function f : G → Γ , we introduce k*random*walks of some length, ℓ say,*on*expander*graphs*to design a probabilistic*homomorphism*test, which could be thought as a*graph*test ...##
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Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
[article]

2015
*
arXiv
*
pre-print

Let H be a fixed

arXiv:1509.06361v2
fatcat:eydgqkabuvhbhdxdxrwarfnqcu
*graph**on*n vertices. Let f_H(G) = 1 iff the input*graph*G*on*n vertices contains H as a (not necessarily induced) subgraph. ... Until very recently, it was believed that the quantum query complexity is at least square root of the*randomized**one*. ... A*homomorphism*from a*graph*H*into*a*graph*G is a function h : V (H) → V (G) such that: if (u, v) ∈ E(H) then (h(u), h(v)) ∈ E(G). ...##
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Distinguishing homomorphisms of infinite graphs
[article]

2013
*
arXiv
*
pre-print

We supply an upper bound

arXiv:1309.0416v1
fatcat:pqohh3phevecvmnzwxehvjdi6a
*on*the distinguishing chromatic number of certain infinite*graphs*satisfying an adjacency property. ... We prove that if a*graph*G satisfies the connected existentially closed property and admits a*homomorphism*to H, then it admits continuum-many distinguishing*homomorphisms*from G to H join K_2. ... The infinite*random**graph*is c.e.c. as it is e.c. The infinite*random*bipartite*graph*is also c.e.c. ...##
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Random graphs of free groups contain surface subgroups
[article]

2013
*
arXiv
*
pre-print

A

arXiv:1303.2700v1
fatcat:7f2kn57ahfbhdiowwaa2esrbva
*random**graph*of free groups contains a surface subgroup ... A fatgraph is a*graph*Y together with a cyclic order*on*the edge incident to each vertex. Lemma 5.1.2. Let φ : F k → F l be a*random**homomorphism*of length n. ... We think of*Z*G as a*graph*without a basepoint. The*graph**Z*G depends only*on*the conjugacy class of G in F . ...##
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Limits of dense graph sequences

2006
*
Journal of combinatorial theory. Series B (Print)
*

Along the way we introduce a rather general model of

doi:10.1016/j.jctb.2006.05.002
fatcat:jzh6p36b5nadhg7tshkuh5ic7m
*random**graphs*, which seems to be interesting*on*its own right. ... We show that if a sequence of dense*graphs*G n has the property that for every fixed*graph*F , the density of copies of F in G n tends to a limit, then there is a natural "limit object," namely a symmetric ... We call G(n, W H ) a*random**graph*with model H . We show that the*homomorphisms*densities*into*G(n, F ) are close to the*homomorphism*densities*into*W . ...##
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Random homomorphisms into the orthogonality graph
[article]

2021
*
arXiv
*
pre-print

The problem can also be formulated as defining and computing

arXiv:2105.03657v1
fatcat:tvtbvz3cmrdinan3xnuow25s3y
*random*orthogonal representations of*graphs*. ... We define subgraph densities in the orthogonality*graphs**on*the unit spheres in dimension d, under appropriate sparsity condition*on*the subgraphs. ... A related question is to define a*random**homomorphism*of G*into*H d . ...##
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Limits of dense graph sequences
[article]

2004
*
arXiv
*
pre-print

Along the lines we introduce a rather general model of

arXiv:math/0408173v2
fatcat:me77hs2tpbdq7ehnn6q5l3grgi
*random**graphs*, which seems to be interesting*on*its own right. ... measurable 2-variable function*on*[0,1]. ... Acknowledgement We are grateful to Jeong-Han Kim for his kind advice*on*Azuma's inequality. ...##
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Some advances on Sidorenko's conjecture

2018
*
Journal of the London Mathematical Society
*

A bipartite

doi:10.1112/jlms.12142
fatcat:c6luggepwzdpvloyjmw6kq6aki
*graph*H is said to have Sidorenko's property if the probability that the uniform*random*mapping from V(H) to the vertex set of any*graph*G is a*homomorphism*is at least the product over all ... First, using branching*random*walks, we develop an embedding algorithm which allows us to prove that bipartite*graphs*admitting a certain type of tree decomposition have Sidorenko's property. ... estimate*on*the entropy of the*random**homomorphism*. ...##
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Algorithmic aspects of M-Lipschitz mappings of graphs
[article]

2018
*
arXiv
*
pre-print

M-Lipschitz mappings of

arXiv:1801.05496v3
fatcat:sijafagz6ngmjmyhtzxk7xwow4
*graphs*(or equivalently*graph*-indexed*random*walks) are a generalization of standard*random*walk*on**Z*. ...*into*a full GI*random*walk for a given*graph*. ... Introduction*Graph*-indexed*random*walks (or also M -Lipschitz mapping of*graphs*) are a generalization of standard*random*walk*on**Z*. ...##
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Signal Processing On Kernel-Based Random Graphs

2018
*
Zenodo
*

The

doi:10.5281/zenodo.1160000
fatcat:l2zooq7mmzfdjia7rkuqu5r5wa
*homomorphism*densities are testable*graph*parameters in that 8✏ > 0 9 an integer k 0 such that for all*graphs*G*on*k > k 0 nodes, a*random*set of vertices X in G satisfies |t(F, G) t(F, G[X])| < ✏ ... Substituting this function*into*the integral in (13)*one*obtains f (x 1 ) = e ( 1 x1+ 0 )*Z*1 0 e (2 1 x2) dx 2 = 1 2 1 (1 e 2 1 )e ( 1 x1+ 0 ) (15) from which it is clear that 1 2 1 (1 e 2 1 )e 0 is the ...
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