scholarly journals A Note on the Glauber Dynamics for Sampling Independent Sets

10.37236/1552 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Eric Vigoda
Keyword(s):  
Maximum Degree ◽  
Mixing Time ◽  
Independent Set ◽  
Glauber Dynamics ◽  
Independent Sets ◽  
General Graphs

This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree $\Delta$. For a positive fugacity $\lambda$, the weight of an independent set $\sigma$ is $\lambda^{|\sigma|}$. Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time $O(n\log{n})$ when $\lambda < {{2}\over {\Delta-2}}$ for triangle-free graphs. We extend their approach to general graphs.

Frozen (Δ + 1)-colourings of bounded degree graphs

2020 ◽  
pp. 1-14
Author(s):  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Guillem Perarnau
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Connected Component ◽  
Bounded Degree ◽  
Large Girth ◽  
Bounded Degree Graphs

Abstract Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

scholarly journals On randomly colouring locally sparse graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Alan Frieze ◽  
Juan Vera
Keyword(s):  
Random Graphs ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Glauber Dynamics ◽  
Average Degree ◽  
Sparse Graphs ◽  
International Audience ◽  
General Graphs

International audience We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ >c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ >α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.

Greedy Algorithms

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Case Finding ◽  
Sequential Algorithm ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Greedy Method ◽  
Numerical Order ◽  
Selection Of

We consider the selection of two basketball teams at a neighborhood playground to illustrate the greedy method. Usually the top two players are designated captains. All other players line up while the captains alternate choosing one player at a time. Usually, the players are picked using a greedy strategy. That is, the captains choose the best unclaimed player. The system of selection of choosing the best, most obvious, or most convenient remaining candidate is called the greedy method. Greedy algorithms often lead to easily implemented efficient sequential solutions to problems. Unfortunately, it also seems to be that sequential greedy algorithms frequently lead to solutions that are inherently sequential — the solutions produced by these algorithms cannot be duplicated rapidly in parallel, unless NC equals P. In the following subsections we will examine this phenomenon. We illustrate some of the important aspects of greedy algorithms using one that constructs a maximal independent set in a graph. An independent set is a set of vertices of a graph that are pairwise nonadjacent. A maximum independent set is such a set of largest cardinality. It is well known that finding maximum independent sets is NP-hard. An independent set is maximal if no other vertex can be added while maintaining the independent set property. In contrast to the maximum case, finding maxima? independent sets is very easy. Figure 7.1.1 depicts a simple polynomial time sequential algorithm computing a maximal independent set. The algorithm is a greedy algorithm: it processes the vertices in numerical order, always attempting to add the lowest numbered vertex that has not yet been tried. The sequential algorithm in Figure 7.1.1, having processed vertices 1,... , j -1, can easily decide whether to include vertex j. However, notice that its decision about j potentially depends on its decisions about all earlier vertices — j will be included in the maximal independent set if and only if all j' less than j and adjacent to it were excluded.

Spanning k-tree with specified vertices

2019 ◽  
Vol 11 (04) ◽  
pp. 1950043
Author(s):  
Feifei Song ◽  
Jianjie Zhou
Keyword(s):  
Maximum Degree ◽  
Spanning Trees ◽  
Independent Set ◽  
Disjoint Paths ◽  
Maximum Independent Set ◽  
Local Connectivity

A [Formula: see text]-tree is a tree with maximum degree at most [Formula: see text]. For a graph [Formula: see text] and [Formula: see text] with [Formula: see text], let [Formula: see text] be the cardinality of a maximum independent set containing [Formula: see text] and [Formula: see text]. For a graph [Formula: see text] and [Formula: see text], the local connectivity [Formula: see text] is defined to be the maximum number of internally disjoint paths connecting [Formula: see text] and [Formula: see text] in [Formula: see text]. In this paper, we prove the following theorem and show the condition is sharp. Let [Formula: see text], [Formula: see text] and [Formula: see text] be integers with [Formula: see text], [Formula: see text] and [Formula: see text]. For any two nonadjacent vertices [Formula: see text] and [Formula: see text] of [Formula: see text], we have [Formula: see text] and [Formula: see text]. Then for any [Formula: see text] distinct vertices of [Formula: see text], [Formula: see text] has a spanning [Formula: see text]-tree such that each of [Formula: see text] specified vertices has degree at most [Formula: see text]. This theorem implies H. Matsuda and H. Matsumura’s result in [on a [Formula: see text]-tree containing specified cleares in a graph, Graphs Combin. 22 (2006) 371–381] and V. Neumann-Lara and E. Rivera-Campo’s result in [Spanning trees with bounded degrees, Combinatorica 11 (1991) 55–61].

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

scholarly journals Hypergraph Independent Sets

2012 ◽  
Vol 22 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JONATHAN CUTLER ◽  
A. J. RADCLIFFE
Keyword(s):  
Upper Bound ◽  
Independent Set ◽  
Independent Sets ◽  
Extremal Problems ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Fixed Size

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

The mixing time of Glauber dynamics for coloring regular trees

2010 ◽  
pp. NA-NA
Author(s):  
Leslie Ann Goldberg ◽  
Mark Jerrum ◽  
Marek Karpinski
Keyword(s):  
Mixing Time ◽  
Glauber Dynamics

Algebraic independence

1981 ◽  
Vol 46 (2) ◽  
pp. 377-384 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Algebraic Closure ◽  
Algebraic Independence ◽  
Independent Set ◽  
Independent Sets ◽  
Large Cardinal ◽  
Similarity Type ◽  
Limit Cardinal ◽  
Free Set ◽  
Infinite Set ◽  
The Universe

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

scholarly journals The independence polynomial of inverse commuting graph of dihedral groups

2020 ◽  
Vol 16 (1) ◽  
pp. 115-120
Author(s):  
Aliyu Suleiman ◽  
Aliyu Ibrahim Kiri
Keyword(s):  
Identity Element ◽  
Independent Set ◽  
Independent Sets ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct  x,y, E D2N, x and y are adjacent if and only if xy = yx = 1  where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

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