scholarly journals New proofs of Konig's bipartite graph characterization theorem

2017 ◽  
Vol 1 (2) ◽  
pp. 32
Author(s):  
Salman Fawzi Ghazal
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Characterization Theorem ◽  
Bipartite Graphs ◽  
Odd Cycles

We introduce four new elementary short proofs of the famous K\"{o}nig's theorem which characterizes bipartite graphs by absence of odd cycles. Our proofs are more elementary than earlier proofs because they use neither distances nor walks nor spanning trees.

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.

scholarly journals On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

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.

scholarly journals Longest cycles in certain bipartite graphs

1998 ◽  
Vol 21 (1) ◽  
pp. 103-106
Author(s):  
Pak-Ken Wong
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Longest Cycles

LetGbe a connected bipartite graph with bipartition(X,Y)such that|X|≥|Y|(≥2),n=|X|andm=|Y|. Suppose, for all verticesx∈Xandy∈Y,dist(x,y)=3impliesd(x)+d(y)≥n+1. ThenGcontains a cycle of length2m. In particular, ifm=n, thenGis hamiltomian.

scholarly journals A Characterization for Sparse $\varepsilon$-Regular Pairs

10.37236/923 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Stefanie Gerke ◽  
Angelika Steger
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Bipartite Graphs ◽  
Regularity Lemma ◽  
Sparse Graphs

We are interested in $(\varepsilon)$-regular bipartite graphs which are the central objects in the regularity lemma of Szemerédi for sparse graphs. A bipartite graph $G=(A\uplus B,E)$ with density $p={|E|}/({|A||B|})$ is $(\varepsilon)$-regular if for all sets $A'\subseteq A$ and $B'\subseteq B$ of size $|A'|\geq \varepsilon|A|$ and $|B'|\geq \varepsilon |B|$, it holds that $\left| {e_G(A',B')}/{(|A'||B'|)}- p\right| \leq \varepsilon p$. In this paper we prove a characterization for $(\varepsilon)$-regularity. That is, we give a set of properties that hold for each $(\varepsilon)$-regular graph, and conversely if the properties of this set hold for a bipartite graph, then the graph is $f(\varepsilon)$-regular for some appropriate function $f$ with $f(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. The properties of this set concern degrees of vertices and common degrees of vertices with sets of size $\Theta(1/p)$ where $p$ is the density of the graph in question.

scholarly journals Biclique edge-choosability in some classes of graphs∗

2017 ◽  
Author(s):  
Gabriel A. G. Sobral ◽  
Marina Groshaus ◽  
André L. P. Guedes
Keyword(s):  
Lower Bound ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
Polynomial Algorithms ◽  
Odd Cycles ◽  
Chordal Bipartite Graphs

In this paper we study the problem of coloring the edges of a graph for any k-list assignment such that there is no maximal monochromatic biclique, in other words, the k-biclique edge-choosability problem. We prove that the K3free graphs that are not odd cycles are 2-star edge-choosable, chordal bipartite graphs are 2-biclique edge-choosable and we present a lower bound for the biclique choice index of power of cycles and power of paths. We also provide polynomial algorithms to compute a 2-biclique (star) edge-coloring for K3-free and chordal bipartite graphs for any given 2-list assignment to edges.

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