3,181 research outputs found
H-Colouring Bipartite Graphs
For graphs G and H, an H-colouring of G (or homomorphism from G to H) is a function from the vertices of G to the vertices of H that preserves adjacency. H-colourings generalize such graph theory notions as proper colourings and independent sets. For a given H, k∈V(H) and G we consider the proportion of vertices of G that get mapped to k in a uniformly chosen H-colouring of G. Our main result concerns this quantity when G is regular and bipartite. We find numbers 0⩽a−(k)⩽a+(k)⩽1 with the property that for all such G, with high probability the proportion is between a−(k) and a+(k), and we give examples where these extremes are achieved. For many H we have a−(k)=a+(k) for all k and so in these cases we obtain a quite precise description of the almost sure appearance of a randomly chosen H-colouring. As a corollary, we show that in a uniform proper q-colouring of a regular bipartite graph, if q is even then with high probability every colour appears on a proportion close to 1/q of the vertices, while if q is odd then with high probability every colour appears on at least a proportion close to 1/(q+1) of the vertices and at most a proportion close to 1/(q−1) of the vertices. Our results generalize to natural models of weighted H-colourings, and also to bipartite graphs which are sufficiently close to regular. As an application of this latter extension we describe the typical structure of H-colourings of graphs which are obtained from n-regular bipartite graphs by percolation, and we show that p=1/n is a threshold function across which the typical structure changes. The approach is through entropy, and extends work of J. Kahn, who considered the size of a randomly chosen independent set of a regular bipartite graph
Self-Assembly of Geometric Space from Random Graphs
We present a Euclidean quantum gravity model in which random graphs
dynamically self-assemble into discrete manifold structures. Concretely, we
consider a statistical model driven by a discretisation of the Euclidean
Einstein-Hilbert action; contrary to previous approaches based on simplicial
complexes and Regge calculus our discretisation is based on the Ollivier
curvature, a coarse analogue of the manifold Ricci curvature defined for
generic graphs. The Ollivier curvature is generally difficult to evaluate due
to its definition in terms of optimal transport theory, but we present a new
exact expression for the Ollivier curvature in a wide class of relevant graphs
purely in terms of the numbers of short cycles at an edge. This result should
be of independent intrinsic interest to network theorists. Action minimising
configurations prove to be cubic complexes up to defects; there are indications
that such defects are dynamically suppressed in the macroscopic limit. Closer
examination of a defect free model shows that certain classical configurations
have a geometric interpretation and discretely approximate vacuum solutions to
the Euclidean Einstein-Hilbert action. Working in a configuration space where
the geometric configurations are stable vacua of the theory, we obtain direct
numerical evidence for the existence of a continuous phase transition; this
makes the model a UV completion of Euclidean Einstein gravity. Notably, this
phase transition implies an area-law for the entropy of emerging geometric
space. Certain vacua of the theory can be interpreted as baby universes; we
find that these configurations appear as stable vacua in a mean field
approximation of our model, but are excluded dynamically whenever the action is
exact indicating the dynamical stability of geometric space. The model is
intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice
Sidorenko's conjecture, colorings and independent sets
Let denote the number of homomorphisms from a graph to a
graph . Sidorenko's conjecture asserts that for any bipartite graph , and
a graph we have where
and denote the number of vertices and edges of the graph and
, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs : for the complete graph on vertices, for a
with a loop added at one of the end vertices, and for a path on vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph . For instance,
for a bipartite graph the number of -colorings
satisfies
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
H-coloring Tori
For graphs G and H, an H-coloring of G is a function from the vertices of G to the vertices of H that preserves adjacency. H-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of H-colorings of the even discrete torus View the MathML source, the graph on vertex set {0,…,m−1}d (m even) with two strings adjacent if they differ by 1 (mod m) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes E and O. In the case m=2 the even discrete torus is the discrete hypercube or Hamming cube Qd, the usual nearest neighbor graph on {0,1}d. We obtain, for any H and fixed m, a structural characterization of the space of H-colorings of View the MathML source. We show that it may be partitioned into an exceptional subset of negligible size (as d grows) and a collection of subsets indexed by certain pairs (A,B)∈V(H)2, with each H-coloring in the subset indexed by (A,B) having all but a vanishing proportion of vertices from E mapped to vertices from A, and all but a vanishing proportion of vertices from O mapped to vertices from B. This implies a long-range correlation phenomenon for uniformly chosen H-colorings of View the MathML source with m fixed and d growing. The special pairs (A,B)∈V(H)2 are characterized by every vertex in A being adjacent to every vertex in B, and having |A||B| maximal subject to this condition. Our main technical result is an upper bound on the probability, for an arbitrary edge uv of View the MathML source, that in a uniformly chosen H-coloring f of View the MathML source the pair View the MathML source is not one of these special pairs (where N⋅ indicates neighborhood). Our proof proceeds through an analysis of the entropy of f, and extends an approach of Kahn, who had considered the case of m=2 and H a doubly infinite path. All our results generalize to a natural weighted model of H-colorings
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