scholarly journals Finite Factors of Bernoulli Schemes and Distinguishing Labelings of Directed Graphs

10.37236/6 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Andrew Lazowski ◽  
Stephen M. Shea
Keyword(s):  
Automorphism Group ◽  
Lower Bound ◽  
Directed Graph ◽  
Directed Graphs ◽  
Bell Numbers ◽  
Undirected Graphs ◽  
Bernoulli Scheme ◽  
Finite Set ◽  
Finite Factor

A labeling of a graph is a function from the vertices of the graph to some finite set.  In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs.  Their definition easily extends to directed graphs.  Let $G$ be a directed graph associated to the $k$-block presentation of a Bernoulli scheme $X$.  We determine the automorphism group of $G$, and thus the distinguishing labelings of $G$.  A labeling of $G$ defines a finite factor of $X$.  We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of $X$.  We use the Bell numbers to find a lower bound for the number of finitarily Markovian finite factors of a Bernoulli scheme.  We show that demarcating labelings of $G$ are distinguishing.

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

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.

VERTEX ISOPERIMETRIC PARAMETER OF A COMPUTATION GRAPH

2012 ◽  
Vol 23 (04) ◽  
pp. 941-964 ◽  
Author(s):  
DESH RANJAN ◽  
MOHAMMAD ZUBAIR
Keyword(s):  
Lower Bound ◽  
Directed Graph ◽  
Complete Solution ◽  
Boundary Vertex ◽  
Undirected Graphs ◽  
Minimum Value ◽  
Hexagonal Shape ◽  
Straight Line ◽  
Financial Application ◽  
Computation Graph

Let G = (V,E) be a computation graph, which is a directed graph representing a straight line computation and S ⊂ V. We say a vertex v is an input vertex for S if there is an edge (v, u) such that v ∉ S and u ∈ S. We say a vertex u is an output vertex for S if there is an edge (u, v) such that u ∈ S and v ∉ S. A vertex is called a boundary vertex for a set S if it is either an input vertex or an output vertex for S. We consider the problem of determining the minimum value of boundary size of S over all sets of size M in an infinite directed grid. This problem is related to the vertex isoperimetric parameter of a graph, and is motivated by the need for deriving a lower bound for memory traffic for a computation graph representing a financial application. We first extend the notion of vertex isoperimetric parameter for undirected graphs to computation graphs, and then provide a complete solution for this problem for all M. In particular, we show that a set S of size M = 3k2 + 3k + 1 vertices of an infinite directed grid, the boundary size must be at least 6k + 3, and this is obtained when the vertices in S are arranged in a regular hexagonal shape with side k + 1.

On the Divisibility of Homogeneous Directed Graphs

1993 ◽  
Vol 45 (2) ◽  
pp. 284-294 ◽  
Author(s):  
M. El-Zahar ◽  
N. W. Sauer
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Isomorphic Copy ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractLet be a finite set of finite tournaments. We will give a necessary and sufficient condition for the -free homogeneous directed graph to be divisible. That is, that there is a partition of into two classes such that neither of them contains an isomorphic copy of .

The Bursty Steiner Tree Problem

2017 ◽  
Vol 28 (07) ◽  
pp. 869-887
Author(s):  
Gokarna Sharma ◽  
Costas Busch
Keyword(s):  
Lower Bound ◽  
Steiner Tree ◽  
Competitive Ratio ◽  
Directed Graphs ◽  
Steiner Tree Problem ◽  
Undirected Graphs ◽  
Tight Bound ◽  
Trade Offs ◽  
Single Burst ◽  
First Results

We introduce and study a new Steiner tree problem variation called the bursty Steiner tree problem where new nodes arrive into bursts. This is an online problem which becomes the well-known online Steiner tree problem if the number of nodes in each burst is exactly one and becomes the classic Steiner tree problem if all the nodes appear in a single burst. In undirected graphs, we provide a tight bound of [Formula: see text] on the competitive ratio for this problem, where [Formula: see text] is the total number of nodes to be connected and [Formula: see text] is the total number of different bursts. In directed graphs of bounded edge asymmetry [Formula: see text], we provide a competitive ratio for this problem with a gap of [Formula: see text] factor between the lower bound and the upper bound. We also show that a tight bound of [Formula: see text] on the competitive ratio can be obtained for a bursty variation of the terminal Steiner tree problem. These are the first results that provide clear performance trade-offs for a novel Steiner tree problem variation that subsumes both of its online and classic versions.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

scholarly journals Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

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