scholarly journals On Cartesian products with small crossing numbers

2012 ◽  
Vol 28 (1) ◽  
pp. 67-75
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Crossing Number ◽  
Necessary And Sufficient Conditions ◽  
Line Graphs ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Necessary And Sufficient

Kulli at al. started to characterize line graphs with crossing number one. In this paper, the similar problems were solved for the Cartesian products of two graphs. The necessary and sufficient conditions are given for all pairs of graphs G1 and G2 for which the crossing number of their Cartesian product G1 × G2 is one or two.

Generation of the alternating group by modular additions

2019 ◽  
Vol 29 (5) ◽  
pp. 303-309
Author(s):  
Fedor M. Malyshev
Keyword(s):  
Permutation Group ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Action ◽  
Alternating Group ◽  
Permutation Groups ◽  
Cartesian Products ◽  
Strong Connectedness ◽  
Necessary And Sufficient

Abstract The paper is concerned with systems of generators of permutation groups on Cartesian products of residue rings. Each separate permutation from the system of generators is constructed on the basis of additions, is characterized by the local action, and leaves fixed the major parts of the components of the element being transformed. A criterion of 2-transitivity of the generated permutation group is given in the form of the strong connectedness of the digraph which corresponds to the system of generators and which is defined on the set of numbers of residue rings in the Cartesian product. Necessary and sufficient conditions under which this group contains an alternating group are formulated.

scholarly journals The crossing number of P 2 5 × Pn

2012 ◽  
Vol 21 (1) ◽  
pp. 65-72
Author(s):  
DANIELA KRAVECOVA ◽  
Keyword(s):  
Cartesian Product ◽  
Crossing Number ◽  
Exact Results ◽  
Cartesian Products ◽  
Crossing Numbers

There are known several exact results concerning crossing numbers of Cartesian products of two graphs. In the paper, we extend these results by giving the crossing number of the Cartesian product ... where Pn is the path of length n and ... is the second power of Pn.

scholarly journals ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH

2019 ◽  
Vol 100 (1) ◽  
pp. 5-12
Author(s):  
MICHAEL HAYTHORPE ◽  
ALEX NEWCOMBE
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Crossing Number ◽  
Star Graph ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
First Time

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

scholarly journals Endomorphism regular Ockham algebras of finite Boolean type

1997 ◽  
Vol 39 (1) ◽  
pp. 99-110 ◽  
Author(s):  
T. S. Blyth ◽  
H. J. Silva
Keyword(s):  
Inverse Semigroup ◽  
Dual Space ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Boolean Lattice ◽  
Necessary And Sufficient Conditions ◽  
Ockham Algebra ◽  
Cyclic Groups ◽  
Ockham Algebras ◽  
Necessary And Sufficient

AbstractIf (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.

The crossing number of G ▭ C n for the graph G on six vertices

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Emília Draženská
Keyword(s):  
Cartesian Product ◽  
Crossing Number ◽  
Cartesian Products ◽  
Crossing Numbers

AbstractThe crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C n for some graphs G on five and six vertices and the cycle C n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C n, where G is a specific graph on six vertices.

scholarly journals Generalized Line Graphs: Cartesian Products and Complexity of Recognition

10.37236/3983 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Aparna Lakshmanan S. ◽  
Csilla Bujtás ◽  
Zsolt Tuza
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
General Setting ◽  
Usual Sense ◽  
Line Graphs ◽  
Free Graph ◽  
Cartesian Products ◽  
Triangle Graph ◽  
Np Complete ◽  
Necessary And Sufficient

Putting the concept of line graph in a more general setting, for a positive integer $k$, the $k$-line graph $L_k(G)$ of a graph $G$ has the $K_k$-subgraphs of $G$ as its vertices, and two vertices of $L_k(G)$ are adjacent if the corresponding copies of $K_k$ in $G$ share $k-1$ vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The $k$-anti-Gallai graph $\triangle_k(G)$ of $G$ is a specified subgraph of $L_k(G)$ in which two vertices are adjacent if the corresponding two $K_k$-subgraphs are contained in a common $K_{k+1}$-subgraph in $G$.We give a unified characterization for nontrivial connected graphs $G$ and $F$ such that the Cartesian product $G\Box F$ is a $k$-line graph. In particular for $k=3$, this answers the question of Bagga (2004), yielding the necessary and sufficient condition that $G$ is the line graph of a triangle-free graph and $F$ is a complete graph (or vice versa). We show that for any $k\ge 3$, the $k$-line graph of a connected graph $G$ is isomorphic to the line graph of $G$ if and only if $G=K_{k+2}$. Furthermore, we prove that the recognition problem of $k$-line graphs and that of $k$-anti-Gallai graphs are NP-complete for each $k\ge 3$.

scholarly journals Metric products and continuation of isotone functions

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
O. Dovgoshey ◽  
E. Petrov ◽  
G. Kozub
Keyword(s):  
Metric Spaces ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Moduli Of Continuity ◽  
Necessary And Sufficient

AbstractLet ℝ+ = [0,∞) and let A ⊆ ℝ+n. We have found the necessary and sufficient conditions under which a function Φ: A → ℝ+ has an isotone subadditive continuation on ℝ+n. It allows us to describe the metrics, defined on the Cartesian product X 1×...×X n of given metric spaces $$\left( {X_1 ,d_{X_1 } } \right), \ldots ,\left( {X_n ,d_{X_n } } \right)$$, generated by the isotone metric preserving functions on ℝ+n. It is also shown that the isotone metric preserving functions Φ: ℝ+n → ℝ+ coincide with the first moduli of continuity of the nonconstant bornologous functions g: ℝ+n → ℝ+.

scholarly journals Quasi-total Graphs with Crossing Numbers

2010 ◽  
Vol 2 (2) ◽  
pp. 257-263
Author(s):  
V. R. Kulli ◽  
B. Basavanagoud ◽  
K. M. Niranjan
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossing Numbers ◽  
Necessary And Sufficient

We establish here necessary and sufficient conditions for quasi-total graphs to have crossing numbers k (k = 1, 2 or 3). Keywords: Quasi-total; Crossing numbers. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.DOI: 10.3329/jsr.v2i2.3527                J. Sci. Res. 2 (2), 257-263 (2010) 

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Sign in / Sign up

Export Citation Format

Share Document