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On Random Graph Homomorphisms into Z
2000
Journal of combinatorial theory. Series B (Print)
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Given a bipartite connected nite 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. ...
doi:10.1006/jctb.1999.1931
fatcat:5prkyt4ktnhohpr3m6gjsol5y4
Random cubic graphs are not homomorphic to the cycle of size 7
2005
Journal of combinatorial theory. Series B (Print)
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We prove that a 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 ...
doi:10.1016/j.jctb.2004.11.004
fatcat:66bmajzgijgabfb4qfhb37iraq
Random cubic graphs are not homomorphic to the cycle of size 7
[article]
2006
arXiv
pre-print
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We prove that a 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 ...
arXiv:math/0701013v1
fatcat:bpxwf3cwwfapjb4mcdhzdt44s4
Removal lemmas and approximate homomorphisms
[article]
2022
arXiv
pre-print
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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. ...
arXiv:2104.11626v2
fatcat:fotjgjbczneyfplbeadibkbble
A cluster expansion approach to exponential random graph models
2012
Journal of Statistical Mechanics: Theory and Experiment
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The exponential family of 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. ...
doi:10.1088/1742-5468/2012/05/p05004
fatcat:r74lpvsjzbdwbpenfq53knk4bu
On the derandomization of the graph test for homomorphism over groups
2011
Theoretical Computer Science
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For abelian groups G = 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 ...
doi:10.1016/j.tcs.2010.12.046
fatcat:zbw6oizap5cetlcbvc5ys7v3za
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
[article]
2015
arXiv
pre-print
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Let H be a fixed 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). ...
arXiv:1509.06361v2
fatcat:eydgqkabuvhbhdxdxrwarfnqcu
Distinguishing homomorphisms of infinite graphs
[article]
2013
arXiv
pre-print
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We supply an upper bound 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. ...
arXiv:1309.0416v1
fatcat:pqohh3phevecvmnzwxehvjdi6a
Random graphs of free groups contain surface subgroups
[article]
2013
arXiv
pre-print
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A 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 . ...
arXiv:1303.2700v1
fatcat:7f2kn57ahfbhdiowwaa2esrbva
Limits of dense graph sequences
2006
Journal of combinatorial theory. Series B (Print)
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Along the way we introduce a rather general model of 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 . ...
doi:10.1016/j.jctb.2006.05.002
fatcat:jzh6p36b5nadhg7tshkuh5ic7m
Random homomorphisms into the orthogonality graph
[article]
2021
arXiv
pre-print
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The problem can also be formulated as defining and computing 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 . ...
arXiv:2105.03657v1
fatcat:tvtbvz3cmrdinan3xnuow25s3y
Limits of dense graph sequences
[article]
2004
arXiv
pre-print
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Along the lines we introduce a rather general model of 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. ...
arXiv:math/0408173v2
fatcat:me77hs2tpbdq7ehnn6q5l3grgi
Some advances on Sidorenko's conjecture
2018
Journal of the London Mathematical Society
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A bipartite 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. ...
doi:10.1112/jlms.12142
fatcat:c6luggepwzdpvloyjmw6kq6aki
Algorithmic aspects of M-Lipschitz mappings of graphs
[article]
2018
arXiv
pre-print
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M-Lipschitz mappings of 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. ...
arXiv:1801.05496v3
fatcat:sijafagz6ngmjmyhtzxk7xwow4
Signal Processing On Kernel-Based Random Graphs
2018
Zenodo
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The 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 ...
doi:10.5281/zenodo.1160000
fatcat:l2zooq7mmzfdjia7rkuqu5r5wa
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