scholarly journals Cycle Decompositions of Complete Digraphs

10.37236/8219 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
A. C. Burgess ◽  
P. Danziger ◽  
M. T. Javed
Keyword(s):  
Directed Graph ◽  
Necessary Conditions ◽  
Complete Solution ◽  
Directed Cycle ◽  
Cycle Decomposition ◽  
Alpha Beta ◽  
Large Cycle ◽  
Cycle Decompositions

In this paper, we consider the problem of decomposing the complete directed graph $K_n^*$ into cycles of given lengths.  We consider general necessary conditions for a directed cycle decomposition of $K_n^*$ into $t$ cycles of lengths $m_1, m_2, \ldots, m_t$  to exist and and provide a powerful construction for creating such decompositions in the case where there is one 'large' cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case.  In particular, when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.

scholarly journals Balanced directed cycle designs based on cyclic groups

1995 ◽  
Vol 58 (2) ◽  
pp. 210-218 ◽  
Author(s):  
Chaufah Nilrat ◽  
Cheryl E. Praeger
Keyword(s):  
Directed Graph ◽  
Cyclic Group ◽  
Complete Classification ◽  
Directed Cycle ◽  
Group Of Automorphisms ◽  
Edge Disjoint ◽  
Cyclic Groups ◽  
Directed Cycles

AbstractA balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

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.

Negative Signed Domination in Digraphs

2011 ◽  
Vol 65 ◽  
pp. 145-147
Author(s):  
Wen Sheng Li
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Domination Number ◽  
Directed Cycle ◽  
Signed Domination ◽  
Signed Domination Number

The Concept of Negative Signed Domination Number of a Directed Graph Is Introduced. Exact Values Are Found for the Directed Cycle and Particular Types of Tournaments. Furthermore, it Is Proved that the Negative Signed Domination Number May Be Arbitrarily Big for Digraphs with a Directed Hamiltonian Cycle.

scholarly journals Directed Graphs Without Short Cycles

2009 ◽  
Vol 19 (2) ◽  
pp. 285-301 ◽  
Author(s):  
JACOB FOX ◽  
PETER KEEVASH ◽  
BENNY SUDAKOV
Keyword(s):  
Directed Graph ◽  
Absolute Constant ◽  
Simple Structure ◽  
Directed Graphs ◽  
Constant Factor ◽  
Directed Cycle ◽  
Strong Component ◽  
Feedback Arc Set ◽  
Directed Cycles

For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G ∖ X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour and Sullivan.This result can also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

Cycle decomposition for the permutations of an infinite set

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Cycle Decomposition ◽  
Infinite Set ◽  
Cycle Decompositions

We present two cycle decompositions for the permutations of an infinite set.

scholarly journals Cycle intersection graphs and minimum decycling sets of even graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050027
Author(s):  
Michael Cary
Keyword(s):  
Upper Bound ◽  
Optimization Problem ◽  
Intersection Graph ◽  
Cycle Decomposition ◽  
Intersection Graphs ◽  
Decycling Number ◽  
Cycle Rank ◽  
Cycle Graph ◽  
Cycle Decompositions

We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization problem over the space of cycle decompositions of even graphs, and we conjecture that the upper bound for the decycling number of even graphs presented in this paper is best possible.

Minus Domination Numbers of Directed Graphs

2011 ◽  
Vol 267 ◽  
pp. 334-337
Author(s):  
Wen Sheng Li ◽  
Hua Ming Xing
Keyword(s):  
Lower Bounds ◽  
Directed Graph ◽  
Maximum Degree ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Domination Number ◽  
Directed Cycle ◽  
Domination Numbers

The concept of minus domination number of an undirected graph is transferred to directed graphs. Exact values are found for the directed cycle and particular types of tournaments. Furthermore, we present some lower bounds for minus domination number in terms of the order, the maximum degree, the maximum outdegree and the minimum outdegree of a directed graph.

scholarly journals The Square of a Directed Graph

2019 ◽  
Vol 8 (4) ◽  
pp. 8331-8335
Keyword(s):  
Directed Graph ◽  
Oriented Graph ◽  
Directed Cycle ◽  
Directed Path ◽  
The Relationship ◽  
Special Case ◽  
Cycle Graph

The square of an oriented graph is an oriented graph such that if and only if for some , both and exist. According to the square of oriented graph conjecture (SOGC), there exists a vertex such that . It is a special case of a more general Seymour’s second neighborhood conjecture (SSNC) which states for every oriented graph , there exists a vertex such that . In this study, the methods to square a directed graph and verify its correctness were introduced. Moreover, some lemmas were introduced to prove some classes of oriented graph including regular oriented graph, directed cycle graph and directed path graphs are satisfied the SOGC. Besides that, the relationship between SOGC and SSNC are also proved in this study. As a result, the verification of the SOGC in turn implies partial results for SSNC.

Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

2019 ◽  
Vol 19 (3) ◽  
pp. 475-494 ◽  
Author(s):  
Lu Chen ◽  
Guozhen Lu ◽  
Chunxia Tao
Keyword(s):  
Half Space ◽  
Hardy Inequality ◽  
Necessary Conditions ◽  
Stereographic Projection ◽  
Pohozaev Identity ◽  
Lagrange Equations ◽  
High Dimensions ◽  
Spherical Form ◽  
Best Constant ◽  
Alpha Beta

Abstract The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space: \int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x% -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|% f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_% {+})} for any nonnegative functions {f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})} , {g\in L^{p}(\partial\mathbb{R}^{n}_{+})} , and {p,q^{\prime}\in(0,1)} , {\beta<\frac{1-n}{p^{\prime}}} or {\alpha<-\frac{n}{q}} , {\lambda>0} satisfying \frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}% =2. Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space {\mathbb{R}^{n}_{+}} .

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