Gibbs rapidly samples colorings of G(n, d/n)

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s–Rényi random graph G(n, d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n, d/n) is d(1?o(1)), it contains many nodes of degree of order (log n) / (log log n). The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n, d/n) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n ^{1+Ω(1 / log log} n) with high probability. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. Almost all known sufficient conditions in terms of number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work we consider sampling q-colorings and show that for every d < ∞ there exists q(d) < ∞ such that for all q ≥ q(d) the mixing time of the Gibbs sampling on G(n, d/n) is polynomial in n with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n, d/n) for d > 1 where the number of colors does not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). In previous work we have shown that similar results hold for the ferromagnetic Ising model. However, the proof for the Ising model crucially relied on monotonicity arguments and the “Weitz tree”, both of which have no counterparts in the coloring setting. Our proof presented here exploits in novel ways the local treelike structure of Erd?s–Rényi random graphs, block dynamics, spatial decay properties and coupling arguments. Our results give the first polynomial-time algorithm to approximately sample colorings on G(n, d/n) with a constant number of colors. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which there exists an α > 0 such that every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub-graph is the union of a tree and at most O(1) edges and where each simple path Γ of length O(log n) satisfies ${\sum_{u \in \Gamma}\sum_{v \neq u}\alpha^{d(u,v)} = O({\rm log} n)}$ . The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.     

Some extremal results in cochromatic and dichromatic theory

For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {G_n} is a family of graphs where G_n has o(n² log²(n)) edges, then z(G_n) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m² ln²(m)).     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

The Swendsen‐Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is conjectured to converge to equilibrium in O(|V|^1/4) steps at any (inverse) temperature β, yet there are few results providing o(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, when β < β_c(d) where β_c(d) denotes the uniqueness/nonuniqueness threshold on infinite d‐regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a monotone version of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW dynamics has mixing time and relaxation time Θ(1) on any graph of maximum degree d for all β < β_c(d). Our proof technology can be applied to general monotone Markov chains, including for example the heat‐bath block dynamics, for which we obtain new tight mixing time bounds.     

A degree sum condition for longest cycles in 3‐connected graphs

For a graph G, we denote by d_G(x) and κ(G) the degree of a vertex x in G and the connectivity of G, respectively. In this article, we show that if G is a 3‐connected graph of order n such that d_G(x) + d_G(y) + d_G(z) ≥ d for every independent set {x, y, z}, then G contains a cycle of length at least min{d ? κ(G), n}. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 277–283, 2007     

1‐Factorizations of pseudorandom graphs

A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E(G). In this paper, we prove that for any ?>0, an (n,d,λ)‐graph G admits a 1‐factorization provided that n is even, C₀ ≤ d ≤ n?1 (where C₀=C₀(?) is a constant depending only on ?), and λ ≤ d^1??. In particular, since (as is well known) a typical random d‐regular graph G_n,d is such a graph, we obtain the existence of a 1‐factorization in a typical G_n,d for all C₀ ≤ d ≤ n?1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1‐factorizations of such graphs G, which is better by a factor of 2^nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

Set Partition Complexes

The Hom complexes were introduced by Lovász to study topological obstructions to graph colorings. The vertices of Hom(G,K _n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Čukić and Kozlov, that Hom(G,K _n ) is (n−d−2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.     

Decreasing the diameter of bounded degree graphs

Let f_d (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n₀ (D) vertices, f₂(G) = n − D − 1 and f₃(G) ≥ n − O(D³). For d ≥ 4, f_d (G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of f_d (G) over all connected graphs on n vertices is n/⌊d/2 ⌋ − O(1). As a byproduct, we show that for the n‐cycle C_n, f_d (C_n) = n/(2⌊d/2 ⌋ − 1) − O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Hadwiger number and chromatic number for near regular degree sequences

We consider a problem related to Hadwiger's Conjecture. Let D=(d₁, d₂, …, d_n) be a graphic sequence with 0?d₁?d₂?···?d_n?n?1. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. Define h(D)=max{h(G): G∈R[D]} and χ(D)=max{χ(G): G∈R[D]}, where h(G) and χ(G) are Hadwiger number and chromatic number of a graph G, respectively. Hadwiger's Conjecture implies that h(D)?χ(D). In this paper, we establish the above inequality for near regular degree sequences. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 175–183, 2010     

The Decomposition Dimension of Graphs

 For an ordered k-decomposition ? = {G ₁, G ₂,…,G _k } of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G ₁), d(e, G ₂),…,d(e, G _k )), where d(e, G _i ) is the distance from e to G _i . A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(K _n ) ≤⌊(2n + 5)/3⌋ for n≥ 3. Received: June 17, 1998 Final version received: August 10, 1999     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

Closure Concepts: A Survey

In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvátal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cl_n (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cl_n(G) is well-defined, and that G is hamiltonian if and only if cl_n(G) is hamiltonian. Moreover, they showed that cl_n(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cl_n(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cl_n(G)=K _n (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvátal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years. Revised: September 27, 1999     

What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Note on a new coloring number of a graph

The distance coloring number X_d(G) of a graph G is the minimum number n such that every vertex of G can be assigned a natural number m ≤ n and no two vertices at distance i are both assigned i. It is proved that for any natural number n there exists a graph G with X_d(G) = n.     

The total chromatic number of regular graphs of high degree

The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) 32 |V (G)| + 263 , where d(G) denotes the degree of a vertex in G, then χT (G) d(G) + 2.     

Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k‐coloring of a random graph G_n,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Gargano,Lewinter and Malerba问题的解

§1　IntroductionLet G be a graph with vertex-set V(G) ={ v1 ,v2 ,...,vn} .A labeling of G is a bijectionL:V(G)→{ 1,2 ,...,n} ,where L (vi) is the label of a vertex vi.A labeled graph is anordered pair (G,L) consisting of a graph G and its labeling L.Definition1.An increasing nonconsecutive path in a labeled graph(G,L) is a path(u1 ,u2 ,...,uk) in G such thatL(ui) + 1相似文献   

On the Wiener Index of Graphs

The Wiener index of a graph G is defined as W(G)=∑ _u,v d _G (u,v), where d _G (u,v) is the distance between u and v in G and the sum goes over all the pairs of vertices. In this paper, we first present the 6 graphs with the first to the sixth smallest Wiener index among all graphs with n vertices and k cut edges and containing a complete subgraph of order n−k; and then we construct a graph with its Wiener index no less than some integer among all graphs with n vertices and k cut edges.