scholarly journals Large Holes in Quasi-Random Graphs

10.37236/784 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Joanna Polcyn
Keyword(s):  
Random Graphs ◽  
Induced Subgraphs ◽  
Degree Condition ◽  
Almost All

Quasi-random graphs have the property that the densities of almost all pairs of large subsets of vertices are similar, and therefore we cannot expect too large empty or complete bipartite induced subgraphs in these graphs. In this paper we answer the question what is the largest possible size of such subgraphs. As an application, a degree condition that guarantees the connection by short paths in quasi-random pairs is stated.

scholarly journals Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

2014 ◽  
Vol 06 (03) ◽  
pp. 1450043
Author(s):  
Bo Ning ◽  
Shenggui Zhang ◽  
Bing Chen
Keyword(s):  
Hamilton Cycles ◽  
Induced Subgraphs ◽  
Degree Sum ◽  
Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

On Random Intersection Graphs: The Subgraph Problem

1999 ◽  
Vol 8 (1-2) ◽  
pp. 131-159 ◽  
Author(s):  
MICHAŁ KAROŃSKI ◽  
EDWARD R. SCHEINERMAN ◽  
KAREN B. SINGER-COHEN
Keyword(s):  
Circuit Design ◽  
Random Graphs ◽  
Induced Subgraphs ◽  
Intersection Graphs ◽  
New Model ◽  
Gate Matrix

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

scholarly journals Vertices cannot be hidden from quantum spatial search for almost all random graphs

2018 ◽  
Vol 17 (4) ◽  
Author(s):  
Adam Glos ◽  
Aleksandra Krawiec ◽  
Ryszard Kukulski ◽  
Zbigniew Puchała
Keyword(s):  
Random Graphs ◽  
Spatial Search ◽  
Almost All

Expected Maximum Block Size in Critical Random Graphs

2019 ◽  
Vol 28 (4) ◽  
pp. 638-655
Author(s):  
V. Rasendrahasina ◽  
A. Rasoanaivo ◽  
V. Ravelomanana
Keyword(s):  
Real Number ◽  
Random Graphs ◽  
Function Theory ◽  
Block Size ◽  
Maximum Size ◽  
Induced Subgraphs ◽  
Symbolic Method ◽  
Exact Analytic Expression ◽  
Analytic Expression ◽  
Image Position

AbstractLet G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.

Achromatic numbers of random graphs

1982 ◽  
Vol 92 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graphs ◽  
Almost All ◽  
Global Invariant

AbstractThe achromatic number ψ(G) of a graph G is the greatest number of colours in a proper colouring of the vertices of G such that for every pair of colours some vertex of the first colour and some vertex of the second colour are adjacent. We prove that almost all graphs Gn with n vertices satisfy n/(k+l) < ψ(Gn) < n/(k–1), where k = k(n) = (log2n)½. We show also that the achromatic number is a ‘global’ invariant.

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