scholarly journals Determining crossing numbers of the join products of two specific graphs of order six with the discrete graph

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2829-2846
Author(s):  
Michal Stas
Keyword(s):  
Connected Graph ◽  
Crossing Number ◽  
Topological Approach ◽  
Crossing Numbers ◽  
Minimum Numbers ◽  
Discrete Graph

The main aim of the paper is to give the crossing number of the join product G* + Dn for the connected graph G* of order six consisting of P4 + D1 and of one leaf incident with some inner vertex of the path P4 on four vertices, and where Dn consists of n isolated vertices. In the proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G* onto the set of subgraphs by which the edges of G* are crossed exactly once. Due to the mentioned algebraic topological approach, we are able to extend known results concerning crossing numbers for join products of new graphs. The proofs are 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 (k-1)! vertices of the graph. Finally, by adding one edge to the graph G*, we are able to obtain the crossing number of the join product of one other graph with the discrete graph Dn.

scholarly journals Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 123 ◽  
Author(s):  
Michal Staš
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
Isolated Vertex ◽  
Minimum Numbers ◽  
Discrete Graph ◽  
Disconnected Graph

The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. 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 D n and with the path P n on n vertices for three other 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.

ON THE ADDITIVITY OF CROSSING NUMBERS OF GRAPHS

2008 ◽  
Vol 17 (09) ◽  
pp. 1043-1050 ◽  
Author(s):  
JESÚS LEAÑOS ◽  
GELASIO SALAZAR
Keyword(s):  
Connected Graph ◽  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Critical Graphs ◽  
Crossing Numbers ◽  
The Individual

We describe a relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G-C. Let G be a connected graph with a 2-edge-cut C := [V1,V2]. Let u1u2, v1v2 be the edges of C, so that ui,vi ∈ Vi for i = 1,2, and let Gi := G[Vi] and G'i := Gi + uivi. We show that if either G1 or G2 is not connected, then cr (G) = cr (G1) + cr (G2), and that if they are both connected then cr (G) = cr (G'1) + cr (G'2). We use this to show how to decompose crossing-critical graphs with 2-edge-cuts into smaller, 3-edge-connected crossing-critical graphs. We also observe that this settles a question arising from knot theory, raised by Sawollek, by describing exactly under which conditions the crossing number of the connected sum of two graphs equals the sum of the crossing numbers of the individual graphs.

DETERMINING CROSSING NUMBERS OF GRAPHS OF ORDER SIX USING CYCLIC PERMUTATIONS

2018 ◽  
Vol 98 (3) ◽  
pp. 353-362 ◽  
Author(s):  
MICHAL STAŠ
Keyword(s):  
Connected Graph ◽  
Crossing Number ◽  
Crossing Numbers

We extend known results concerning crossing numbers by giving the crossing number of the join product$G+D_{n}$, where the connected graph$G$consists of one$4$-cycle and of two leaves incident with the same vertex of the$4$-cycle, and$D_{n}$consists of$n$isolated vertices. The proofs are done with the help of software that generates all cyclic permutations for a given number$k$and creates a graph for calculating the distances between all$(k-1)!$vertices of the graph.

scholarly journals On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 135
Author(s):  
Michal Staš
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
The Individual ◽  
Discrete Graph

In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + D n for the graph G ∗ on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables.

scholarly journals Cyclic permutations and crossing numbers of join products of symmetric graph of order six

2018 ◽  
Vol 34 (2) ◽  
pp. 143-155
Author(s):  
STEFAN BEREZNY ◽  
◽  
MICHAL STAS ◽  
Keyword(s):  
Crossing Number ◽  
Symmetric Graph ◽  
Combinatorial Properties ◽  
Crossing Numbers ◽  
Join Of Graphs ◽  
Discrete Graph

In the paper, we extend 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 two leaves incident with two opposite vertices of one 4-cycle, 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 a calculating the distances between all (k − 1)! vertices of the graph. Finally, by adding new edges to the graph G, we are able to obtain the crossing number of the join product with the discrete graph Dn for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

scholarly journals The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1277
Author(s):  
Michal Staš
Keyword(s):  
Connected Graph ◽  
Crossing Numbers ◽  
Tripartite Graph ◽  
Complete Tripartite Graph

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

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 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 Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

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