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 Digraphs with eulerian chains

1986 ◽  
Vol 41 (3) ◽  
pp. 298-303
Author(s):  
D. K. Skilton
Keyword(s):  
Directed Graph ◽  
Undirected Graphs ◽  
Survey Paper ◽  
Similar Work

AbstractAn eulerian chain in a directed graph is a continuous directed route which traces every arc of the digraph exactly once. Such a route may be finite or infinite, and may have 0, 1 or 2 end vertices. For each kind of eulerian chain, there is a characterization of those diagraphs possessing such a route. In this survey paper we strealine these characterizations, and then synthesize them into a single description of all digraphs having some eulerian chain. Similar work has been done for eulerian chains in undirected graphs, so we are able to compare corresponding results for graphs and digraphs.

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 On star coloring of Mycielskians

2018 ◽  
Vol 2 (2) ◽  
pp. 82
Author(s):  
K. Kaliraj ◽  
V. Kowsalya ◽  
Vernold Vivin
Keyword(s):  
Chromatic Number ◽  
Graph Transformation ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Complete Bipartite Graphs ◽  
Star Coloring

<p>In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph <span class="math"><em>G</em></span> into a new graph <span class="math"><em>μ</em>(<em>G</em>)</span>, we now call the Mycielskian of <span class="math"><em>G</em></span>, which has the same clique number as <span class="math"><em>G</em></span> and whose chromatic number equals <span class="math"><em>χ</em>(<em>G</em>) + 1</span>. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.</p>

A Minimax Equality Related to the Longest Directed Path in an Acyclic Graph

1975 ◽  
Vol 27 (2) ◽  
pp. 348-351 ◽  
Author(s):  
K. Vidyasankar ◽  
D. H. Younger
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Path ◽  
Acyclic Graph ◽  
Vertex Set ◽  
Acyclic Directed Graph

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG

The general position problem and strong resolving graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1126-1135 ◽  
Author(s):  
Sandi Klavžar ◽  
Ismael G. Yero
Keyword(s):  
Connected Graph ◽  
General Position ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Direct Products ◽  
Strong Product ◽  
Product Graphs ◽  
Complete Bipartite Graphs ◽  
Position Problem

Abstract The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(GSR), where GSR is the strong resolving graph of G, and ω(GSR) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp(G ⊠ H) ≥ gp(G)gp(H), and asked whether the equality holds for arbitrary connected graphs G and H. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

The maximum 1-2 matching problem and two kinds of its variants

2019 ◽  
Vol 11 (05) ◽  
pp. 1950050
Author(s):  
Hiroki Izumi ◽  
Yuki Nishida ◽  
Sennosuke Watanabe ◽  
Yoshihide Watanabe
Keyword(s):  
Bipartite Graphs ◽  
The Other ◽  
Undirected Graphs ◽  
Matching Problem

We present an algorithm for solving the maximum 1-2 matching problem in bipartite graphs by reducing it to solving the 1-1 matching in general undirected graphs. Further, we propose two variants of the maximum 1-2 matching problem: one is the maximum 1-2 matching problem of the different kind in divided bipartite graphs and the other is the maximum 1-2 matching problem of the same kind in divided bipartite graphs. The former one is shown to be solved by a similar algorithm to that of the maximum 1-2 matching problem by reducing it to solve the maximum 1-1 matching problem in bipartite graphs. For the latter one, we can only present the augmenting subgraph theorem, which is an extension of our augmenting trail theorem for the maximum 1-2 matching problem.

On the Line Degree Splitting Graph of a Graph

2017 ◽  
Vol 18 ◽  
pp. 1-10
Author(s):  
B. Basavanagoud ◽  
Roopa S. Kusugal
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Splitting Graph

In this paper, we introduce the concept of the line degree splitting graph of a graph. We obtain some properties of this graph. We find the girth of the line degree splitting graphs. Further, we establish the characterization of graphs whose line degree splitting graphs are eulerian, complete bipartite graphs and complete graphs.

scholarly journals Inertia Sets of Semicliqued Graphs

2021 ◽  
Vol 37 ◽  
pp. 747-757
Author(s):  
Amy Yielding ◽  
Taylor Hunt ◽  
Joel Jacobs ◽  
Jazmine Juarez ◽  
Taylor Rhoton ◽  
...  
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Undirected Graphs ◽  
Minimum Rank ◽  
Original Graph ◽  
Complete Bipartite Graphs ◽  
Special Classes

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.

A very rapid in situ Kikuchi technique to determine relative crystal misorientation

1988 ◽  
Vol 46 ◽  
pp. 830-831
Author(s):  
J. I. Bennetch
Keyword(s):  
Electron Beam ◽  
Analytical Techniques ◽  
Superplastic Forming ◽  
The Other ◽  
Kikuchi Pattern ◽  
Kikuchi Line ◽  
Line Analysis

In a recent study of the superplastic forming (SPF) behavior of certain Al-Li-X alloys, the relative misorientation between adjacent (sub)grains proved to be an important parameter. It is well established that the most accurate way to determine misorientation across boundaries is by Kikuchi line analysis. However, the SPF study required the characterization of a large number of (sub)grains in each sample to be statistically meaningful, a very time-consuming task even for comparatively rapid Kikuchi analytical techniques.In order to circumvent this problem, an alternate, even more rapid in-situ Kikuchi technique was devised, eliminating the need for the developing of negatives and any subsequent measurements on photographic plates. All that is required is a double tilt low backlash goniometer capable of tilting ± 45° in one axis and ± 30° in the other axis. The procedure is as follows. While viewing the microscope screen, one merely tilts the specimen until a standard recognizable reference Kikuchi pattern is centered, making sure, at the same time, that the focused electron beam remains on the (sub)grain in question.

Rapid Isolation of High Molecular Weight Urokinase from Native Human Urine

1982 ◽  
Vol 47 (03) ◽  
pp. 197-202 ◽  
Author(s):  
Kurt Huber ◽  
Johannes Kirchheimer ◽  
Bernd R Binder
Keyword(s):  
Molecular Weight ◽  
High Molecular Weight ◽  
Human Urine ◽  
Gel Filtration ◽  
Chain Structure ◽  
The Other ◽  
Single Chain ◽  
Apparent Homogeneity ◽  
Sequential Adsorption

SummaryUrokinase (UK) could be purified to apparent homogeneity starting from crude urine by sequential adsorption and elution of the enzyme to gelatine-Sepharose and agmatine-Sepharose followed by gel filtration on Sephadex G-150. The purified product exhibited characteristics of the high molecular weight urokinase (HMW-UK) but did contain two distinct entities, one of which exhibited a two chain structure as reported for the HMW-UK while the other one exhibited an apparent single chain structure. The purification described is rapid and simple and results in an enzyme with probably no major alterations. Yields are high enough to obtain purified enzymes for characterization of UK from individual donors.

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