scholarly journals Vertex-Disjoint Subtournaments of Prescribed Minimum Outdegree or Minimum Semidegree: Proof for Tournaments of a Conjecture of Stiebi

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Nicolas Lichiardopol
Keyword(s):  
Maximum Order ◽  
Strongly Connected ◽  
Vertex Disjoint ◽  
Directed Cycles

It was proved (Bessy et al., 2010) that for r≥1, a tournament with minimum semidegree at least 2r−1 contains at least r vertex-disjoint directed triangles. It was also proved (Lichiardopol, 2010) that for integers q≥3 and r≥1, every tournament with minimum semidegree at least (q−1)r−1 contains at least r vertex-disjoint directed cycles of length q. None information was given on these directed cycles. In this paper, we fill a little this gap. Namely, we prove that for d≥1 and r≥1, every tournament of minimum outdegree at least ((d2+3d+2)/2)r−(d2+d+2)/2 contains at least r vertex-disjoint strongly connected subtournaments of minimum outdegree d. Next, we prove for tournaments a conjecture of Stiebitz stating that for integers s≥1 and t≥1, there exists a least number f(s,t) such that every digraph with minimum outdegree at least f(s,t) can be vertex-partitioned into two sets inducing subdigraphs with minimum outdegree at least s and at least t, respectively. Similar results related to the semidegree will be given. All these results are consequences of two results concerning the maximum order of a tournament of minimum outdegree d (of minimum semidegree d) not containing proper subtournaments of minimum outdegree d (of minimum semidegree d).

scholarly journals Disjoint Cycles of Different Lengths in Graphs and Digraphs

10.37236/6921 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Julien Bensmail ◽  
Ararat Harutyunyan ◽  
Ngoc Khang Le ◽  
Binlong Li ◽  
Nicolas Lichiardopol
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Probabilistic Method ◽  
Undirected Graphs ◽  
Disjoint Cycles ◽  
Vertex Disjoint ◽  
Degree Bound ◽  
Directed Cycles

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

A Note on p-Cyclic Matrices and Digraphs

1967 ◽  
Vol 10 (4) ◽  
pp. 497-501
Author(s):  
B. R. Heap ◽  
M. S Lynn
Keyword(s):  
Adjacency Matrix ◽  
Greatest Common Divisor ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Directed Cycles

We use the terminology of [1]. Let D be a strongly connected digraph on n points and containing m lines, and let A = A(D) be the corresponding adjacency matrix, so that A is an n x n 0-1 matrix containing m unit elements. We recall that A and D are said to be p-cyclic if p is the greatest common divisor of the lengths of all directed cycles of D. Clearly, the larger the value of p, the smaller the value of m must be; in this note we make the latter and related statements precise.

Intimation

2017 ◽  
Author(s):  
G. R. F. Ferrari
Keyword(s):  
Social Life ◽  
Relevance Theory ◽  
Self Presentation ◽  
Erving Goffman ◽  
Strongly Connected ◽  
Dan Sperber ◽  
Paul Grice

The communicative scale is introduced. What is fundamental to communication is the intention of the communicator rather than the codes that languages employ. Following the model first proposed by Paul Grice and developed in Dan Sperber and Deirdre Wilson’s ‘relevance theory’, the structure of communicative intentionality is understood to be recursive: its underlying form is ‘I want you to know that I want you to know’. This leaves room for a simpler kind of transmission, to be called ‘intimation’, whose underlying form would be ‘I want you to know’. If communication is a transmission at the ‘full-on’ position of the scale, and if the switch is off when no communication is intended, then intimation would be at the intermediate, ‘half-on’ position. Intimation is particularly useful in contexts where discretion, suggestiveness, or plausible deniability are needed. It is strongly connected to self-presentation in social life (as studied by Erving Goffman).

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

scholarly journals A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

2021 ◽  
Author(s):  
Vera Traub ◽  
Thorben Tröbst
Keyword(s):  
Vehicle Routing Problem ◽  
Covering Problem ◽  
Post Processing ◽  
Routing Problem ◽  
Cycle Cover ◽  
Disjoint Cycles ◽  
Processing Step ◽  
Number Of Cycles ◽  
Vertex Disjoint ◽  
Capacitated Vehicle

AbstractWe consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.

scholarly journals The Nexus of Sophisticated Digital Assets with Economic Policy Uncertainty: A Survey of Empirical Findings and an Empirical Investigation

2021 ◽  
Vol 13 (10) ◽  
pp. 5383
Author(s):  
Nikolaos A. Kyriazis
Keyword(s):  
Economic Policy ◽  
Empirical Investigation ◽  
Econometric Analysis ◽  
Policy Uncertainty ◽  
Negative Effects ◽  
Economic Policy Uncertainty ◽  
Strongly Connected ◽  
Threshold Arch ◽  
Best Fit ◽  
Digital Assets

This paper sets out to explore the nexus between economic policy uncertainty (EPU) and digital currencies. An integrated survey takes place based on eleven primary studies. Furthermore, an econometric analysis is conducted by the threshold ARCH, simple asymmetric ARCH and non-linear ARCH specifications covering the bull and the bear markets as well as the highly volatile period up to the present. Threshold ARCH is found to provide the best fit for estimations. Outcomes reveal that Bitcoin is strongly connected with EPU while Ethereum and Litecoin are not but are strongly linked with Bitcoin performance. Moreover, weak negative effects of the VIX on both cryptocurrencies are detected while oil exerts weak positive impacts on Ethereum. Overall, Ethereum and Litecoin could serve for diversifiers against Bitcoin or hedgers against traditional assets during highly stressed periods with the advantage of not being affected by economic policy uncertainty news.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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