scholarly journals Computing the number of h-edge spanning forests in complete bipartite graphs

2014 ◽  
Vol Vol. 16 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Rebecca Stones
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Analysis Of Algorithms ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Spanning Forests ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Number Of Spanning Trees

Analysis of Algorithms International audience Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m ≤50 and n ≤50 in under 2 days.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
Author(s):  
GUILLEM PERARNAU ◽  
ORIOL SERRA
Keyword(s):  
Bipartite Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Complete Bipartite Graphs ◽  
Edge Colourings ◽  
Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

Finite locally-primitive complete bipartite graphs

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Wenwen Fan ◽  
Cai Heng Li ◽  
Jiangmin Pan
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Complete Bipartite Graphs

Abstract.We characterize groups which act locally-primitively on a complete bipartite graph. The result particularly determines certain interesting factorizations of groups.

Note on T-Minimal Complete Bipartite Graphs

1968 ◽  
Vol 11 (5) ◽  
pp. 729-732 ◽  
Author(s):  
I. Z. Bouwer ◽  
I. Broere
Keyword(s):  
Natural Number ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Different Types ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

The thickness of a graph G is the smallest natural number t such that G is the union of t planar subgraphs. A graph G is t-minimal if its thickness is t and if every proper subgraph of G has thickness < t. (These terms were introduced by Tutte in [3]. In [1, p. 51] Beineke employs the term t-critical instead of t-minimal.) The complete bipartite graph K(m, n) consists of m 'dark1 points, n 'light' points, and the mn lines joining points of different types.

scholarly journals The IC-Indices of Complete Bipartite Graphs

10.37236/767 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Chin-Lin Shiue ◽  
Hung-Lin Fu
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Connected Subgraph ◽  
Maximum Value ◽  
Complete Bipartite Graphs ◽  
Function Mapping

Let $G$ be a connected graph, and let $f$ be a function mapping $V(G)$ into ${\Bbb N}$. We define $f(H)=\sum_{v\in{V(H)}}f(v)$ for each subgraph $H$ of $G$. The function $f$ is called an IC-coloring of $G$ if for each integer $k$ in the set $\{1,2,\cdots,f(G)\}$ there exists an (induced) connected subgraph $H$ of $G$ such that $f(H)=k$, and the IC-index of $G$, $M(G)$, is the maximum value of $f(G)$ where $f$ is an IC-coloring of $G$. In this paper, we show that $M(K_{m,n})=3\cdot2^{m+n-2}-2^{m-2}+2$ for each complete bipartite graph $K_{m,n},\,2\leq m\leq n$.

scholarly journals ALGORITHMS FOR CONSTRUCTING EDGE MAGIC TOTAL LABELING OF COMPLETE BIPARTITE GRAPHS

2015 ◽  
pp. 55-58
Author(s):  
KRISHNAPPA H. K ◽  
N K. SRINATH ◽  
S. Manjunath ◽  
RAMAKANTH KUMAR P
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Graph Labeling ◽  
Constant Independent ◽  
Magic Square ◽  
Complete Bipartite Graphs ◽  
One To One ◽  
Definition Of

The study of graph labeling has focused on finding classes of graphs which admits a particular type of labeling. In this paper we consider a particular class of graphs which demonstrates Edge Magic Total Labeling. The class we considered here is a complete bipartite graph Km,n. There are various graph labeling techniques that generalize the idea of a magic square has been proposed earlier. The definition of a magic labeling on a graph with v vertices and e edges is a one to one map taking the vertices and edges onto the integers 1,2,3,………, v+e with the property that the sum of the label on an edge and the labels of its endpoints is constant independent of the choice of edge. We use m x n matrix to construct edge magic total labeling of Km,n.

scholarly journals Number of Spanning Trees of Different Products of Complete and Complete Bipartite Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-23 ◽  
Author(s):  
S. N. Daoud
Keyword(s):  
Tensor Product ◽  
Linear Algebra ◽  
Spanning Trees ◽  
Matrix Theory ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Normal Product ◽  
Complete Bipartite Graphs ◽  
Composition Product ◽  
Number Of Spanning Trees

Spanning trees have been found to be structures of paramount importance in both theoretical and practical problems. In this paper we derive new formulas for the complexity, number of spanning trees, of some products of complete and complete bipartite graphs such as Cartesian product, normal product, composition product, tensor product, symmetric product, and strong sum, using linear algebra and matrix theory techniques.

scholarly journals Results on a Chromatic Number of a Bi-polar Fuzzy Complete Bipartite Graphs and Labelling of Tri-Polar Fuzzy Graphs

2021 ◽  
Vol 12 (2) ◽  
pp. 1040-1046
Author(s):  
Remala Mounika Lakshmi, Et. al.
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
Research Work ◽  
Bipartite Graphs ◽  
Network System ◽  
Route Network ◽  
Fuzzy Graphs ◽  
Complete Bipartite Graphs ◽  
Ultimate Objective

The ultimate objective of a piece of research work is to present the labelling of vertices in 3-PFG and labelling of distances in 3-PFG. Also, we characterize some of its properties. Later, we define the vertex and edge chromatic number BF- Complete Bipartite graph. Further we illustrated an example for BFRGS which represents a Route Network system.

scholarly journals Asymptotic enumeration on self-similar graphs with two boundary vertices

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
General Setting ◽  
Scaling Factor ◽  
Asymptotic Enumeration ◽  
Spanning Forests ◽  
International Audience ◽  
Self Similar ◽  
Connected Subgraphs ◽  
Graph Parameters ◽  
Number Of Spanning Trees

Combinatorics International audience We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.

scholarly journals A Quantitative Approach to Perfect One-Factorizations of Complete Bipartite Graphs

10.37236/4122 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Natacha Astromujoff ◽  
Martin Matamala
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Quantitative Approach ◽  
Hamiltonian Cycles ◽  
Complete Bipartite Graphs

Given a one-factorization $\mathcal{F}$ of the complete bipartite graph $K_{n,n}$, let ${\sf pf}(\mathcal{F})$ denote the number of Hamiltonian cycles obtained by taking pairwise unions of perfect matchings in $\mathcal{F}$. Let ${\sf pf}(n)$ be the maximum of ${\sf pf}(\mathcal{F})$ over all one-factorizations $\mathcal{F}$ of $K_{n,n}$. In this work we prove that ${\sf pf}(n)\geq n^2/4$, for all $n\geq 2$.

Sign in / Sign up

Export Citation Format

Share Document