THE TENEVA GAME

2012 ◽  
Vol 21 (14) ◽  
pp. 1250125 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
PEDRO LOPES
Keyword(s):  
Upper Bound ◽  
Previous Article ◽  
Torus Knot ◽  
Minimum Number ◽  
Modulo P ◽  
Rational Knots ◽  
Future Work

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

scholarly journals Intrinsic linking and knotting in tournaments

2019 ◽  
Vol 28 (12) ◽  
pp. 1950076
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Upper Bound ◽  
Directed Edge ◽  
Undirected Graphs ◽  
Oriented Cycle ◽  
Minimum Number ◽  
The Difference

A directed graph [Formula: see text] is intrinsically linked if every embedding of that graph contains a nonsplit link [Formula: see text], where each component of [Formula: see text] is a consistently oriented cycle in [Formula: see text]. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ([Formula: see text]), intrinsically knotted ([Formula: see text]), intrinsically 3-linked ([Formula: see text]), intrinsically 4-linked ([Formula: see text]), intrinsically 5-linked ([Formula: see text]), intrinsically [Formula: see text]-linked ([Formula: see text]), intrinsically linked with knotted components ([Formula: see text]), and the disjoint linking property ([Formula: see text]). We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic [Formula: see text]-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in [Formula: see text], and provide an upper bound at each [Formula: see text].

scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals The Chromatic Number of Finite Group Cayley Tables

10.37236/7874 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Kevin Halasz ◽  
E. S. Mahmoodian
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Chromatic Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Latin Square ◽  
Finite Abelian Groups ◽  
Cayley Table ◽  
Minimum Number ◽  
Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

scholarly journals Exploring the binding properties and structural stability of an opsin in the chytrid Spizellomyces punctatus using comparative and molecular modeling

2017 ◽  
Author(s):  
Steven R Ahrendt ◽  
Edgar Mauricio Medina ◽  
Chia-en A Chang ◽  
Jason E Stajich
Keyword(s):  
Molecular Modeling ◽  
Structural Characteristics ◽  
Sequence Similarity ◽  
Ligand Docking ◽  
Binding Properties ◽  
Function Relationship ◽  
Relationship Of ◽  
Future Work

Background. Opsin proteins are seven transmembrane receptor proteins which detect light. Opsins can be classified into two types and share little sequence identity: type 1, typically found in bacteria, and type 2, primarily characterized in metazoa. The type 2 opsins (Rhodopsins) are a subfamily of G-protein coupled receptors (GPCRs), a large and diverse class of seven transmembrane proteins and are generally restricted to metazoan lineages. Fungi use light receptors including opsins to sense the environment and transduce signals for developmental or metabolic changes. Opsins characterized in the Dikarya (Ascomycetes and Basidiomycetes) are of the type 1 bacteriorhodopsin family but the early diverging fungal lineages have not been as well surveyed. We identified by sequence similarity a rhodopsin-like GPCR in genomes of early diverging chytrids and examined the structural characteristics of this protein to assess its likelihood to be homologous to animal rhodopsins and bind similar chromophores. Methods. We used template-based structure modeling, automated ligand docking, and molecular modeling to assess the structural and binding properties of an identified opsin-like protein found in Spizellomyces punctatus, a unicellular, flagellated species belonging to Chytridiomycota, one of the earliest diverging fungal lineages. We tested if sequence and inferred structure were consistent with a solved crystal structure of a type 2 rhodopsin from the squid Todarodes pacificus. Results. Our results indicate that the Spizellomyces opsin has structural characteristics consistent with functional animal type 2 rhodopsins and is capable of maintaining a stable structure when associated with the retinaldehyde chromophore, specifically the 9-cis­-retinal isomer. Together, these results support further the homology of Spizellomyces opsins to animal type 2 rhodopsins. Discussion. This represents the first test of structure/function relationship of a type 2 rhodopsin identified in early branching fungal lineages, and provides a foundation for future work exploring pathways and components of photoreception in early fungi.

scholarly journals Exploring the binding properties and structural stability of an opsin in the chytridSpizellomyces punctatususing comparative and molecular modeling

PeerJ ◽  
2017 ◽  
Vol 5 ◽  
pp. e3206 ◽  
Author(s):  
Steven R. Ahrendt ◽  
Edgar Mauricio Medina ◽  
Chia-en A. Chang ◽  
Jason E. Stajich
Keyword(s):  
Molecular Modeling ◽  
Structural Characteristics ◽  
Sequence Similarity ◽  
Ligand Docking ◽  
Binding Properties ◽  
Function Relationship ◽  
Relationship Of ◽  
Future Work

BackgroundOpsin proteins are seven transmembrane receptor proteins which detect light. Opsins can be classified into two types and share little sequence identity: type 1, typically found in bacteria, and type 2, primarily characterized in metazoa. The type 2 opsins (Rhodopsins) are a subfamily of G-protein coupled receptors (GPCRs), a large and diverse class of seven transmembrane proteins and are generally restricted to metazoan lineages. Fungi use light receptors including opsins to sense the environment and transduce signals for developmental or metabolic changes. Opsins characterized in the Dikarya (Ascomycetes and Basidiomycetes) are of the type 1 bacteriorhodopsin family but the early diverging fungal lineages have not been as well surveyed. We identified by sequence similarity a rhodopsin-like GPCR in genomes of early diverging chytrids and examined the structural characteristics of this protein to assess its likelihood to be homologous to animal rhodopsins and bind similar chromophores.MethodsWe used template-based structure modeling, automated ligand docking, and molecular modeling to assess the structural and binding properties of an identified opsin-like protein found inSpizellomyces punctatus, a unicellular, flagellated species belonging to Chytridiomycota, one of the earliest diverging fungal lineages. We tested if the sequence and inferred structure were consistent with a solved crystal structure of a type 2 rhodopsin from the squidTodarodes pacificus.ResultsOur results indicate that theSpizellomycesopsin has structural characteristics consistent with functional animal type 2 rhodopsins and is capable of maintaining a stable structure when associated with the retinaldehyde chromophore, specifically the 9-cis-retinal isomer. Together, these results support further the homology ofSpizellomycesopsins to animal type 2 rhodopsins.DiscussionThis represents the first test of structure/function relationship of a type 2 rhodopsin identified in early branching fungal lineages, and provides a foundation for future work exploring pathways and components of photoreception in early fungi.

scholarly journals Exploring the binding properties and structural stability of an opsin in the chytrid Spizellomyces punctatus using comparative and molecular modeling

2016 ◽  
Author(s):  
Steven R Ahrendt ◽  
Edgar Mauricio Medina ◽  
Chia-en A Chang ◽  
Jason E Stajich
Keyword(s):  
Molecular Modeling ◽  
Structural Characteristics ◽  
Sequence Similarity ◽  
Ligand Docking ◽  
Binding Properties ◽  
Function Relationship ◽  
Relationship Of ◽  
Future Work

Background. Opsin proteins are seven transmembrane receptor proteins which detect light. Opsins can be classified into two types and share little sequence identity: type 1, typically found in bacteria, and type 2, primarily characterized in metazoa. The type 2 opsins (Rhodopsins) are a subfamily of G-protein coupled receptors (GPCRs), a large and diverse class of seven transmembrane proteins and are generally restricted to metazoan lineages. Fungi use light receptors including opsins to sense the environment and transduce signals for developmental or metabolic changes. Opsins characterized in the Dikarya (Ascomycetes and Basidiomycetes) are of the type 1 bacteriorhodopsin family but the early diverging fungal lineages have not been as well surveyed. We identified by sequence similarity a rhodopsin-like GPCR in genomes of early diverging chytrids and examined the structural characteristics of this protein to assess its likelihood to be homologous to animal rhodopsins and bind similar chromophores. Methods. We used template-based structure modeling, automated ligand docking, and molecular modeling to assess the structural and binding properties of an identified opsin-like protein found in Spizellomyces punctatus, a unicellular, flagellated species belonging to Chytridiomycota, one of the earliest diverging fungal lineages. We tested if sequence and inferred structure were consistent with a solved crystal structure of a type 2 rhodopsin from the squid Todarodes pacificus. Results. Our results indicate that the Spizellomyces opsin has structural characteristics consistent with functional animal type 2 rhodopsins and is capable of maintaining a stable structure when associated with the retinaldehyde chromophore, specifically the 9-cis­-retinal isomer. Together, these results support further the homology of Spizellomyces opsins to animal type 2 rhodopsins. Discussion. This represents the first test of structure/function relationship of a type 2 rhodopsin identified in early branching fungal lineages, and provides a foundation for future work exploring pathways and components of photoreception in early fungi.

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