scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

2019 ◽  
Vol 36 (2) ◽  
pp. 205-220
Author(s):  
Bernardo Ábrego ◽  
Silvia Fernández-Merchant ◽  
Athena Sparks
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

THE LINEAR 6-ARBORICITY OF THE COMPLETE BIPARTITE GRAPH Km,n

2013 ◽  
Vol 05 (04) ◽  
pp. 1350029
Author(s):  
SHENGJIE HE ◽  
LIANCUI ZUO
Keyword(s):  
Bipartite Graph ◽  
Undirected Graph ◽  
Complete Bipartite Graph ◽  
Linear Arboricity ◽  
Minimum Number

A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity of G, denote by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In the case where the lengths of paths are not restricted, we then have the linear arboricity of G which is denoted by la(G). In this paper, we obtain some results about the linear 6-arboricity of the complete bipartite graph Km,n.

scholarly journals On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2441
Author(s):  
Michal Staš
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
The Family ◽  
Minimum Number ◽  
The Common ◽  
First Time ◽  
Edge Crossings

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

scholarly journals The Linear 2- and 4-Arboricity of Complete Bipartite Graph Km,n

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Liancui Zuo ◽  
Bing Xue ◽  
Shengjie He
Keyword(s):  
Bipartite Graph ◽  
Undirected Graph ◽  
Complete Bipartite Graph ◽  
Linear Arboricity ◽  
Minimum Number

A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity of G, denoted by lak(G), is the minimum number of linear k-forests needed to decompose G. In case the lengths of paths are not restricted, we then have the linear arboricity of G, denoted by la(G). In this paper, the exact value of the linear 2- and 4-arboricity of complete bipartite graph Km,n for some m and n is obtained.

scholarly journals On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}

2021 ◽  
Vol 41 (1) ◽  
pp. 95-112
Author(s):  
Michal Staš ◽  
Juraj Valiska
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Minimum Number ◽  
Edge Crossings

The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).

scholarly journals The Rectilinear Crossing Number of $K_{10}$ is $62$

10.37236/1567 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Alex Brodsky ◽  
Stephane Durocher ◽  
Ellen Gethner
Keyword(s):  
Crossing Number ◽  
Line Segments ◽  
Combinatorial Argument ◽  
Straight Line ◽  
Minimum Number ◽  
Edge Crossings

The rectilinear crossing number of a graph $G$ is the minimum number of edge crossings that can occur in any drawing of $G$ in which the edges are straight line segments and no three vertices are collinear. This number has been known for $G=K_n$ if $n \leq 9$. Using a combinatorial argument we show that for $n=10$ the number is 62.

scholarly journals On a Theorem of Erdős, Rubin, and Taylor on Choosability of Complete Bipartite Graphs

10.37236/1670 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Alexandr Kostochka
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Ordinary Sense ◽  
Minimum Order ◽  
Minimum Number ◽  
Complete Bipartite Graphs

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not $r$-choosable and the minimum number of edges in an $r$-uniform hypergraph that is not $2$-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete $k$-partite graphs and complete $k$-uniform $k$-partite hypergraphs.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling
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