scholarly journals A tight Erdős-Pósa function for planar minors

2019 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Tony Huyn ◽  
Gwenaël Joret ◽  
Jean-Florent Raymond
Keyword(s):  
Planar Graph ◽  
General Theorem ◽  
Large Body ◽  
Minimum Size ◽  
Property A ◽  
Disjoint Cycles ◽  
The Family ◽  
Odd Cycle ◽  
A Minor ◽  
Odd Cycles

Let F be a family of graphs. Then for every graph G the maximum number of disjoint subgraphs of G, each isomorphic to a member of F, is at most the minimum size of a set of vertices that intersects every subgraph of G isomorphic to a member of F. We say that F packs if equality holds for every graph G. Only very few families pack. As the next best weakening we say that F has the Erdős-Pósa property if there exists a function f such that for every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic to a member of F or a set of at most f(k) vertices that intersects every subgraph of G isomorphic to a member of F. The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph G and integer k>0 the graph G has either k disjoint cycles or a set of O(klogk) vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with f(k)=O(klogk). In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer ℓ, a sufficiently large Escher Wall has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least ℓ times. In particular, it contains no set of ℓ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles. By now there is a large body of literature proving that various families F have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph H the family F(H) of all graphs with a minor isomorphic to H has the Erdős-Pósa property. (When H is non-planar, F(H) does not have the Erdős-Pósa property.) The present paper proves that for every planar graph H the family F(H) has the Erdős-Pósa property with f(k)=O(klogk), which is asymptotically best possible for every graph H with at least one cycle.

Right to Parental Care

2017 ◽  
Author(s):  
Asha Bajpai
Keyword(s):  
Law Reform ◽  
Best Interest ◽  
Family Courts ◽  
Custody Of Children ◽  
The Family ◽  
Shared Parenting ◽  
The Right ◽  
A Minor ◽  
And Control ◽  
Physical Care

Custody refers to the physical care and control of a minor whereas guardianship is a wider term and includes rights and duties with respect to the care and control of minor’s person and property, and includes the right to make decisions relating to the minor. The present legal regime relating to guardianship and custody of children is discussed, including the Guardians and Wards Act, 1890, the Hindu Minority and Guardianship Act, 1956, the personal and matrimonial laws, and relevant provisions in the Family Courts Act and Protection of Women against Domestic Violence Act, 2005. The emerging concepts of shared parenting, joint custody, and the interparental child removal or abduction of child is included. There is review and analysis of some major reported judicial decisions. A comparative survey of international laws and trends has been done. Suggestions for law reform in the best interest of the child have been given.

The Oxford Handbook of IPOs

2018 ◽  
Keyword(s):  
United States ◽  
Initial Public Offerings ◽  
Financial Regulation ◽  
Large Body ◽  
Public Information ◽  
The United States ◽  
Minimum Size ◽  
Stock Exchanges ◽  
Public Offerings ◽  
Firm’S Life Cycle

Firms generally begin as privately owned entities. When they grow large enough, the decision to go public and its consequences are among the most crucial times in a firm’s life cycle. The first time a firm is a reporting issuer gives rise to tremendous responsibilities about disclosing public information and accountability to a wide array of retail shareholders and institutional investors. Initial public offerings (IPOs) offer tremendous opportunities to raise capital. The economic and legal landscape for IPOs has been rapidly evolving across countries. There have been fewer IPOs in the United States in the aftermath of the 2007–2009 financial crisis and associated regulatory reforms that began in 2002. In 1980–2000, an average of 310 firms went public every year, while in 2001–2014 an average of 110 firms went public every year. At the same time, there are so many firms that seek an IPO in China that there has been a massive waiting list of hundreds of firms in recent years. Some countries are promoting small junior stock exchanges to go public early, and even crowdfunding to avoid any prospectus disclosure. Financial regulation of analysts and investment banks has been evolving in ways that drastically impact the economics of going public—in some countries, such as the United States, drastically increasing the minimum size of a company before it can expect to go public. This Handbook not only systematically and comprehensively consolidates a large body of literature on IPOs, but provides a foundation for future debates and inquiry.

Sense of local identity, attitudes toward dialects and language teaching: The Hungarian minority in Serbia

2018 ◽  
Vol 54 (1) ◽  
pp. 115-146 ◽  
Author(s):  
Eleonora Kovacs Rac ◽  
Sabina Halupka-Rešetar
Keyword(s):  
Ethnic Group ◽  
Rural Areas ◽  
Mother Tongue ◽  
Large Body ◽  
Value Judgments ◽  
Language Retention ◽  
Language Varieties ◽  
The Family ◽  
8Th Grade ◽  
Identity Attitudes

Abstract A large body of academic literature (e.g. Fishman 1977, 1999; Giles and Johnson 1981; Romaine 2000, among others) claims that language is one of the most significant markers of ethnic identification and that it plays a crucial role not only in the external perception of an ethnic group by outsiders but also in the selfidentification of an ethnic group. In a minority environment, sense of ethnic identity and language retention are connected very tightly, which is why it is of extreme importance to study attitudes towards the dialects of a language and value judgments about them. The paper presents the results of a research into attitudes toward dialects, conducted with approximately three hundred 5th and 8th grade pupils (age 12 and 15, respectively) attending school in Hungarian in two regions of Vojvodina, Serbia. It explores the subjects’ local features of identity, given that the research was conducted in eight different localities. The results of the research serve as a sound basis for developing use-centered, functional-situational mother tongue education of Hungarian minority pupils living in Serbia, since the current curriculum completely disregards the language varieties of many Hungarian minority pupils brought up and living in rural areas, who acquire and use the dialect spoken in the family.

scholarly journals Legal Peculiarities of Child Custody of Minor Parents with an Incomplete Parental Status

2020 ◽  
Vol 6 ◽  
pp. 22-25
Author(s):  
Ekaterina E. Lekanova ◽  
Keyword(s):  
Russian Federation ◽  
Legal Status ◽  
Legal Protection ◽  
Legal Regulation ◽  
Minor Parent ◽  
The Family ◽  
The Russian Federation ◽  
A Minor ◽  
The Guardian ◽  
Family Code

Despite the existence of an article in modern Russian legislation on the legal status of minor parents, many legal issues related to the implementation and protection of the rights, duties, interests of a minor parent and his child remained outside family legal regulation, which exacerbates the already difficult problem of legal protection of early parenthood. Moreover, the provisions of Article 62 of the Family Code of the Russian Federation are very inharmoniously combined with the rules of guardianship of minors. The aim of the work is to analyze the legislation on the legal status of minor parents and guardians, to identify the legal characteristics of the care of a child of minor parents. The author concludes that the features of the care of a child of minor parents, in addition to the age of one or both parents, in the case of the appointment of a guardian include: a combination of parenthood and guardianship; unequal opportunities for the care of a child by a minor parent who is not able to independently provide care, and by the legal representative of the child of the minor parent; special (additional) grounds for terminating guardianship of a child of minor parents; the need for the guardian to live together not only with the child in care, but also with his parent. The norms of paragraph 2 of article 62 of the Family Code of the Russian Federation and paragraph 2 of article 29 of the Federal law «On Guardianship and Custodianship» should be adjusted. It is proposed to introduce special rules for the selection of the guardian of a child of a minor parent, which would properly ensure the right of the minor parent to live together with the child.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

scholarly journals Tournaments as Feedback Arc Sets

10.37236/1214 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Garth Isaak
Keyword(s):  
Linear Programming ◽  
Lower Bounds ◽  
Integer Linear Programming ◽  
Minimum Size ◽  
Programming Formulation ◽  
Binary Expansion ◽  
Feedback Arc Set ◽  
Disjoint Cycles ◽  
Linear Programming Formulation ◽  
Integer Linear Programming Formulation

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

scholarly journals Ramsey Numbers of Trees Versus Odd Cycles

10.37236/5731 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Matthew Brennan
Keyword(s):  
Positive Integer ◽  
Constant Factor ◽  
Linear Functions ◽  
Ramsey Numbers ◽  
Odd Cycle ◽  
Odd Cycles

Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.

scholarly journals Vertex-Transitive Direct Products of Graphs

10.37236/6999 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Wilfried Imrich
Keyword(s):  
Direct Product ◽  
Direct Products ◽  
Odd Cycle ◽  
Vertex Transitive ◽  
Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

scholarly journals On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

10.37236/5173 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Special Class ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Approach ◽  
The Family ◽  
Edge Colourings ◽  
Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

A new species of Pergalumna (Acari: Oribatida: Galumnidae) from southeastern Iran, including a key to all species of the genus from the Palaearctic region

2021 ◽  
Vol 26 (2) ◽  
pp. 379-394
Author(s):  
Mohammad Ali Akrami ◽  
Fatemeh Ordouni ◽  
Sara Ramroodi
Keyword(s):  
New Species ◽  
Body Size ◽  
Large Body ◽  
Oribatid Mite ◽  
Palaearctic Region ◽  
The Family ◽  
Surface Ornamentation ◽  
Porose Area ◽  
A New Species ◽  
Median Pore

A new species of oribatid mite of the family Galumnidae, Pergalumna sistanbaluchestanica sp. nov., is described and illustrated based on adult specimens from soil in Sistan and Baluchestan province, southeastern Iran. The new species is characterized by dentate rostrum; long interlamellar setae; long, setiform, finely barbed bothridial setae; complete dorsosejugal furrow; large, nearly triangular porose areas Aa; presence of median pore in females and males; large, elongated postanal porose area and large body size. The new species is most similar morphologically to P. seminervosa Mahunka & Mahunka-Papp, 2008, however, differs from it in morphology of bothridial setae and notogastral porose areas Aa and also by the surface ornamentation of the pteromorphs and genital plates; localization of setal alveoli la and lyrifissures im and body size. An identification key to known species of Pergalumna from the Palaearctic region is given.

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