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 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.

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.

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 On the crossing numbers of Cartesian products of paths with special graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 317-325
Author(s):  
MARIAN KLESC ◽  
◽  
DANIELA KRAVECOVA ◽  
JANA PETRILLOVA ◽  
◽  
...  
Keyword(s):  
Cartesian Product ◽  
Exact Results ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Crossing Numbers

There are known exact results of the crossing numbers of the Cartesian product of all graphs of order at most four with paths, cycles and stars. Moreover, for the path Pn of length n, the crossing numbers of Cartesian products GPn for all connected graphs G on five vertices and for forty graphs G on six vertices are known. In this paper, we extend these results by determining the crossing numbers of the Cartesian products GPn for six other graphs G of order six.

scholarly journals Minimal number of crossings in strong product of paths

2013 ◽  
Vol 29 (1) ◽  
pp. 27-32
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
MATUS VALO ◽  
◽  
...  
Keyword(s):  
Crossing Number ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Strong Product ◽  
Product Of Graphs

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. The exact crossing number is known only for few specific families of graphs. Cartesian products of two graphs belong to the first families of graphs for which the crossing number has been studied. Some results concerning crossing numbers are also known for join products of two graphs. In the paper, we start to collect the crossing numbers for the strong product of graphs, namely for the strong product of two paths.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

2019 ◽  
Vol 35 (2) ◽  
pp. 137-146
Author(s):  
STEFAN BEREZNY ◽  
MICHAL STAS ◽  
◽  
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Discrete Graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

scholarly journals ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

2016 ◽  
Vol 4 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lower Bound ◽  
Probability Distributions ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Entropy Theory ◽  
Reverse Inequality ◽  
Cartesian Products ◽  
Model Spaces ◽  
Sofic Entropy ◽  
Sofic Groups

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

On the decomposition of spaces in Cartesian products and unions

1959 ◽  
Vol 55 (3) ◽  
pp. 248-256 ◽  
Author(s):  
Tudor Ganea ◽  
Peter J. Hilton
Keyword(s):  
Topological Space ◽  
Homotopy Type ◽  
General Problem ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Contractible Spaces

The present paper is concerned with particular cases, obtained by suitably restricting the spaces involved, of the following general problem.Given a topological space X, we ask whether there exist integers n ≥ 2 and non-contractible spaces X1, …, Xn such that X has the homotopy type of the Cartesian product X1, × … × Xn or of the union X1, v … v Xn.

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