Zero-one laws for graphs with edge probabilities decaying with distance. Part II

Saharon Shelah

Displaying similar documents to “Zero-one laws for graphs with edge probabilities decaying with distance. Part II”

Zero-one laws for graphs with edge probabilities decaying with distance. Part I

Saharon Shelah (2002)

Fundamenta Mathematicae

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Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{| i - j |} = 1 / {| i - j |}^{α}$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

Chvátal's Condition cannot hold for both a graph and its complement

Alexandr V. Kostochka, Douglas B. West (2006)

Discussiones Mathematicae Graph Theory

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Chvátal’s Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d₁, ...,dₙ in nondecreasing order, i < n/2 implies that $d_{i} > i$ or $d_{n - i} \geq n - i$ . We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2.

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

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We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

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A $(0, 2)$ -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0, λ)$ -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0, 2)$ -graph. $(0, 2)$ -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0, λ)$ -graphs and more specifically...

Acyclic reducible bounds for outerplanar graphs

Mieczysław Borowiecki, Anna Fiedorowicz, Mariusz Hałuszczak (2009)

Discussiones Mathematicae Graph Theory

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For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G [V_{i}] \in_{i}$ for i = 1,...,n, 2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that $u \in V_{i}$ and $v \in V_{j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring....

Existence of graphs with sub exponential transitions probability decay and applications

Clément Rau (2010)

Bulletin de la Société Mathématique de France

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In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp (- n^{α}),$ for fixed $α \in [0, 1 [$ and with Følner function $exp (n^{\frac{2 α}{1 - α}})$ . This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random...

Graceful signed graphs

Mukti Acharya, Tarkeshwar Singh (2004)

Czechoslovak Mathematical Journal

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A $(p, q)$ -sigraph $S$ is an ordered pair $(G, s)$ where $G = (V, E)$ is a $(p, q)$ -graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$ , respectively, where $m + n = q$ . Given positive integers $k$ and $d$ , $S$ is said to be $(k, d)$ -graceful if the vertices of $G$ can be labeled with distinct integers from the set ${0, 1, \dots, k + (q - 1) d}$ such that when each edge $u v$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to...

On generalized shift graphs

Christian Avart, Tomasz Łuczak, Vojtěch Rödl (2014)

Fundamenta Mathematicae

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In 1968 Erdős and Hajnal introduced shift graphs as graphs whose vertices are the k-element subsets of [n] = 1,...,n (or of an infinite cardinal κ ) and with two k-sets $A = a ₁, . . ., a_{k}$ and $B = b ₁, . . ., b_{k}$ joined if $a ₁ < a ₂ = b ₁ < a ₃ = b ₂ < \dots < a_{k} = b_{k - 1} < b_{k}$ . They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with...

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let $G$ be an undirected simple connected graph, and $e = u v$ be an edge of $G$ . Let $N_{G} (e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$ . Let $𝒩_{e}$ be the class of all graphs $H$ such that, for some graph $G$ , $N_{G} (e) ≅ H$ for every edge $e$ of $G$ . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $𝒩_{e}$ . Balasubramanian and Alsardary [1] obtained some other graphs in $𝒩_{e}$ . In this paper we given some new graphs in $𝒩_{e}$ .

Uniquely partitionable graphs

Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G [V_{i}]$ induced by $V_{i}$ has property $_{i}$ ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property...

Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

Izak Broere, Jozef Bucko, Peter Mihók (2002)

Discussiones Mathematicae Graph Theory

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Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G [V_{i}] \in_{i}$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_{i}$ and $_{j}$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

Clusters in middle-phase percolation on hyperbolic plane

Jan Czajkowski (2011)

Banach Center Publications

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I consider p-Bernoulli bond percolation on transitive, nonamenable, planar graphs with one end and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i.e. $0 < p_{c} (G) < p_{u} (G) < 1$ , where $p_{c}$ is the critical probability and $p_{u}$ -the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in ∂ℍ². This result is similar to some results in [Lal].

Nearly complete graphs decomposable into large induced matchings and their applications

Noga Alon, Ankur Moitra, Benjamin Sudakov (2013)

Journal of the European Mathematical Society

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We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $(\frac{N}{2}) - o (N^{2})$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1 - o (1)}$ . The second construction provides a covering of all edges of the complete graph $K_{N}$ by two graphs, each being the edge disjoint union of at most $N^{2 - δ}$ induced matchings, where $δ > 0, 076$ . This disproves (in a strong form) a conjecture of Meshulam,...

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

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We characterize the class [...] L32 $L_{3}^{2}$ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_{3}^{2}$ in the class of threshold graphs, where n is the number of vertices of a tested graph.