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 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 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 .

scholarly journals LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

World Science ◽  
2021 ◽  
Author(s):  
Albert Khachik Sahakyan
Keyword(s):  
Bipartite Graph ◽  
Polynomial Algorithm ◽  
Undirected Graph ◽  
Complete Bipartite Graph ◽  
Vertex Coloring ◽  
List Coloring ◽  
Block Graph ◽  
Block Graphs ◽  
Complete Bipartite Graphs ◽  
Np Complete

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

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

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 Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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