The independent domination number of a random graph

Lane Clark; Darin Johnson

Displaying similar documents to “The independent domination number of a random graph”

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.

Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Christophe Sabot, Pierre Tarrès (2015)

Journal of the European Mathematical Society

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Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph $G$ and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time....

Iterated neighborhood graphs

Martin Sonntag, Hanns-Martin Teichert (2012)

Discussiones Mathematicae Graph Theory

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The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph $(V, E_{N})$ where $E_{N}$ = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph $N^{k} (G) : = N (N (. . . N (G)))$ of G. In particular we investigate conditions for G and k such that $N^{k} (G)$ becomes a complete graph.

Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci (2001)

Discussiones Mathematicae Graph Theory

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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In the case where G is a claw-free graph, G* is equal to G². We define $σ ° ₜ = m i n \sum_{x \in S} d_{G} (x) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t$ . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

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We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam, T. Asir (2012)

Discussiones Mathematicae Graph Theory

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In this paper, we consider the intersection graph $I_{Γ} (ℤ ₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ} (ℤ ₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ} (ℤ ₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ} (ℤ ₙ)$ .

Coherent randomness tests and computing the $K$ -trivial sets

Laurent Bienvenu, Noam Greenberg, Antonín Kučera, André Nies, Dan Turetsky (2016)

Journal of the European Mathematical Society

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We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all $K$ -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$ -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis,...

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..

Potentially H-bigraphic sequences

Michael Ferrara, Michael Jacobson, John Schmitt, Mark Siggers (2009)

Discussiones Mathematicae Graph Theory

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We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of...

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ be a simple graph. A $3$ -valued function $f V (G) \to {- 1, 0, 1}$ is said to be a minus dominating function if for every vertex $v \in V$ , $f (N [v]) = \sum_{u \in N [v]} f (u) \geq 1$ , where $N [v]$ is the closed neighborhood of $v$ . The weight of a minus dominating function $f$ on $G$ is $f (V) = \sum_{v \in V} f (v)$ . The minus domination number of a graph $G$ , denoted by $γ^{-} (G)$ , equals the minimum weight of a minus dominating function on $G$ . In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$ , then $γ^{-} (G) \geq 4 (\sqrt{n + 1} - 1) - n .$ (2) For any negative integer $k$ and any positive integer...

Tail and moment estimates for sums of independent random variables with logarithmically concave tails

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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For random variables $S = \sum_{i = 1}^{\infty} α_{i} ξ_{i}$ , where $(ξ_{i})$ is a sequence of symmetric, independent, identically distributed random variables such that $l n P (| ξ_{i} | \geq t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

On the limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

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We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $(T^{'})$ . We show that for every $ϵ > 0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp (- {(log n)}^{d - ϵ})$ . This bound almost matches the known lower bound of $exp (- C {(log n)}^{d})$ , and signiﬁcantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched...

Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

Rafał M. Łochowski (2006)

Banach Center Publications

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Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S = \sum a_{i ₁, . . ., i_{d}} X_{i ₁}^{(1)} \dots X_{i_{d}}^{(d)}$ generated by symmetric random variables $X_{i ₁}^{(1)}, . . ., X_{i_{d}}^{(d)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments...

Collisions of random walks

Martin T. Barlow, Yuval Peres, Perla Sousi (2012)

Annales de l'I.H.P. Probabilités et statistiques

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A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$ , started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $ℤ^{d}$ with $d \geq 19$ and the uniform spanning tree in $ℤ^{2}$ all have the infinite collision property. For power-law combs and spherically...

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_{r} (G)$ , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_{r} (U) \geq ⎡ n / 3 ⎤$ , and provide a characterization of graphs achieving this bound.

Excited against the tide: a random walk with competing drifts

Mark Holmes (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$ th cookie at a site determines the transition probabilities (to the left and right) for the $k$ th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_{f}$ of all finite undirected graphs. If G is a graph from $C_{f}$ , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

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Let ${X_{n, j}, 1 \leq j \leq m (n), n \geq 1}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ of the dependent bootstrap samples ${{X_{n, j}^{*}, 1 \leq j \leq m (n)}, n \geq 1}$ arising from a sequence of i.i.d. random variables ${X_{n}, n \geq 1}$ .