The Thompson chain of subgroups of\break the Conway~group Co1 and complete graphs on ${n}$~vertices

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Robert T. Curtis
Keyword(s):  
Simple Group ◽  
Finite Family ◽  
The Other ◽  
Complete Graphs ◽  
Directed Edge

AbstractThe large Conway simple groupRemarkably, we can start at the other end in the sense that if we considerSpecifically, we associate with each directed edgeThus this is not simply a sequence of nested subgroups in a larger group, but a finite family of closely-related perfect groups.

Proper Coloring Distance in Edge-Colored Cartesian Products of Complete Graphs and Cycles

2019 ◽  
Vol 29 (04) ◽  
pp. 1950016
Author(s):  
Ajay Arora ◽  
Eddie Cheng ◽  
Colton Magnant
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Complete Graphs ◽  
Specific Class ◽  
Proper Coloring ◽  
Cartesian Products ◽  
General Graph ◽  
Proper Distance ◽  
Cyclic Graphs

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

scholarly journals A Construction for the Hat Problem on a Directed Graph

10.37236/1994 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Rani Hod ◽  
Marcin Krzywkowski
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Maximum Clique ◽  
Clique Number ◽  
Bipartite Graphs ◽  
The Other ◽  
Complete Graphs ◽  
Undirected Graphs ◽  
Asymptotically Optimal

A team of $n$ players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (04) ◽  
pp. 910-920
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λ E , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ E of eavesdropper nodes, there exists a critical intensity λ c < ∞ such that for all λ > λ c the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

scholarly journals On Symmetry of Complete Graphs over Quadratic and Cubic Residues

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
M. Haris Mateen ◽  
M. Khalid Mahmood ◽  
Shahbaz Ali ◽  
M. D. Ashraful Alam
Keyword(s):  
Positive Integer ◽  
Quadratic Residue ◽  
The Other ◽  
Complete Graphs

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod   n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i    d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod   n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

scholarly journals Popular Matchings in Complete Graphs

Algorithmica ◽  
2021 ◽  
Author(s):  
Ágnes Cseh ◽  
Telikepalli Kavitha
Keyword(s):  
Complete Graph ◽  
The Other ◽  
Complete Graphs ◽  
Matching Problem ◽  
Graph Theoretic ◽  
The Given ◽  
Popular Matching

AbstractOur input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { M , M ′ } in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes $$\texttt {NP}$$ NP -complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and $$\texttt {NP}$$ NP -complete when n has the other parity.

scholarly journals ON CONJUGACY CLASSES OF THE KLEIN SIMPLE GROUP IN CREMONA GROUP

2016 ◽  
Vol 59 (2) ◽  
pp. 395-400
Author(s):  
HAMID AHMADINEZHAD
Keyword(s):  
Simple Group ◽  
Three Dimensional ◽  
Conjugacy Classes ◽  
The Other ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Cremona Group ◽  
Del Pezzo

AbstractWe consider countably many three-dimensional PSL2($\mathbb{F}$7)-del Pezzo surface fibrations over ℙ1. Conjecturally, they are all irrational except two families, one of which is the product of a del Pezzo surface with ℙ1. We show that the other model is PSL2($\mathbb{F}$7)-equivariantly birational to ℙ2×ℙ1. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group Cr3(ℂ).

scholarly journals Piercing Axis-Parallel Boxes

10.37236/7034 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Maria Chudnovsky ◽  
Sophie Spirkl ◽  
Shira Zerbib
Keyword(s):  
Pairwise Disjoint ◽  
Finite Family ◽  
The Other ◽  
Axis Parallel ◽  
Special Case

Let $\mathcal{F}$ be a finite family of axis-parallel boxes in $\mathbb{R}^d$ such that $\mathcal{F}$ contains no $k+1$ pairwise disjoint boxes. We prove that if $\mathcal{F}$ contains a subfamily $\mathcal{M}$ of $k$ pairwise disjoint boxes with the property that for every $F\in \mathcal{F}$ and $M\in \mathcal{M}$ with $F \cap M \neq \emptyset$, either $F$ contains a corner of $M$ or $M$ contains $2^{d-1}$ corners of $F$, then $\mathcal{F}$ can be pierced by $O(k)$ points. One consequence of this result is that if $d=2$ and the ratio between any of the side lengths of any box is bounded by a constant, then $\mathcal{F}$ can be pierced by $O(k)$ points. We further show that if for each two intersecting boxes in $\mathcal{F}$ a corner of one is contained in the other, then $\mathcal{F}$ can be pierced by at most $O(k\log\log(k))$ points, and in the special case where $\mathcal{F}$ contains only cubes this bound improves to $O(k)$.

Youth Savings

2018 ◽  
pp. 94-104
Author(s):  
Dean Karlan ◽  
Jacob Appel
Keyword(s):  
Simple Group ◽  
Financial Literacy ◽  
The Other ◽  
Literacy Curriculum ◽  
Technical Design ◽  
Local Partner ◽  
Cover Up ◽  
The Church ◽  
Over Time

This chapter focuses on a study which assesses two sets of policy levers that can be pushed to increase savings: improving the accounts people can access and encouraging people to save more. The researchers worked with local partner organizations on two interventions: a simple group-based savings account and a youth-focused financial literacy curriculum made up of ten 90-minute sessions, to be held weekly for ten weeks. They partnered with the Church of Uganda, whose network of youth clubs counted thousands of members across the country. Keeping track of individuals proved far more difficult than calling names, as club members attempted to cover up each other's absences. This case qualifies as a failure of technical design. Given a setting where official documents are scarce, identifying and tracking individuals over time is always a challenge. The other candidate for a failure of technical design is the set of underlying incentives that led club members to behave badly.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (4) ◽  
pp. 910-920 ◽  
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

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