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### 2-factors in dense bipartite graphs

2002 Discrete Mathematics
n=2 + 1, then G contains an n-ladder.  ...  An n-ladder is a balanced bipartite graph with vertex sets A = {a; : : : ; an} and B = {b1; : : : ; bn} such that ai ∼ bj i |i − j| 6 1.  ...  In Section 3 we review the regularity and blow-up lemmas. In the remainder of this section, we review our notation and work out some easy details of the theory of dense bipartite graphs.  ...

### Magic and supermagic dense bipartite graphs

Jaroslav Ivančo
2007 Discussiones Mathematicae Graph Theory
In the paper we prove that any balanced bipartite graph with minimum degree greater than |V (G)|/4 ≥ 2 is magic. A similar result is presented for supermagic regular bipartite graphs.  ...  A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex  ...  Magic Graphs In this section we show that dense balanced bipartite graphs are magic.  ...

### Efficient Exact Algorithms for Maximum Balanced Biclique Search in Bipartite Graphs [article]

Lu Chen, Chengfei Liu, Rui Zhou, Jiajie Xu, Jianxin Li
2020 arXiv   pre-print
This algorithm in fact can find an MBB in near polynomial time for dense bipartite graphs that are common for applications such as VLSI design.  ...  We propose two exact algorithms dedicated for dense and sparse bipartite graphs respectively. For dense bipartite graphs, an O^*( 1.3803^n) algorithm is proposed.  ...  Novel algorithm for dense bipartite graphs . We observe that bipartite graphs are quite dense in applications such as VLSI design.  ...

### PERS: A Personalized and Explainable POI Recommender System [article]

Ramesh Baral, Tao Li
2017 arXiv   pre-print
user-aspect category bipartite relation as a bipartite graph, and models the explainable recommendation using bipartite core-based and ranking-based methods.  ...  ., Facebook) have many factors (for instance, ratings, check-in time, etc.) that play a crucial role for the Point-of-Interest (POI) recommendations.  ...  be modeled by extracting the dense subgraphs from the bipartite graph.  ...

### Separation Choosability and Dense Bipartite Induced Subgraphs

Louis Esperet, Ross Kang, Stéphan Thomassé
2019 Combinatorics, probability & computing
For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?  ...  We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon.  ...  to 2 in Theorem 3.5.  ...

### 1-factorizations of pseudorandom graphs [article]

Asaf Ferber, Vishesh Jain
2018 arXiv   pre-print
This lower bound is better by a factor of 2^nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.  ...  Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G which is off by a factor of 2 in the base of the exponent from the known upper bound.  ...  Every regular balanced bipartite graph has a 1-factorization. In fact, as the following theorem due to Schrijver [31] shows, a regular balanced bipartite graph has many 1-factorizations. d!  ...

### Progressive Algorithms for Domination and Independence

Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science
, such as powers of nowhere dense classes, map graphs, and (for r = 1) biclique-free graphs.  ...  We demonstrate our approach by giving linear-time fixed-parameter algorithms for the Distance-r Dominating Set problem (parameterized by the solution size) in a wide variety of restricted graph classes  ...  polynomial factor in the running time bound.  ...

### A better tester for bipartiteness? [article]

Andrej Bogdanov, Fan Li
2010 arXiv   pre-print
Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ϵ-far from bipartite, then the subgraph induced by a random subset of O(1/ϵ) vertices is bipartite with high probability.  ...  As it is known that non-adaptive testers for bipartiteness require Ω(1/ϵ^2) queries (Bogdanov and Trevisan, CCC 2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness  ...  A graph is ε-far from bipartite (in the dense graph model) if it is ε n 2 -removed from bipartite. For a graph G and a subset of vertices S, we write G S for the induced subgraph of G on S.  ...

### Matching through Embedding in Dense Graphs [article]

Nitish K. Panigrahy, Prithwish Basu, Don Towsley
2020 arXiv   pre-print
Finding optimal matchings in dense graphs is of general interest and of particular importance in social, transportation and biological networks.  ...  In this paper, we propose a novel network embedding based heuristic algorithm to solve various matching problems in dense graphs.  ...  International Traffic in Arms Regulations or the U.S. Export Administration Regulations.  ...

### DENSE GRAPHLET STATISTICS OF PROTEIN INTERACTION AND RANDOM NETWORKS

R. COLAK, F. HORMOZDIARI, F. MOSER, A. SCHÖNHUTH, J. HOLMAN, M. ESTER, S. C. SAHINALP
2008 Biocomputing 2009
However, there is abundant evidence that in PPI networks modules can be identified with highly interconnected (dense) and/or bipartite subgraphs.  ...  We also prove that another class of emulators, the low-dimensional geometric random graphs (GRGs) cannot contain a specific type of motifs, complete bipartite graphs, which are abundant in PPI networks  ...  A K n,m is a complete, bipartite graph such that |V 1 | = n, |V 2 | = m.  ...

### On the diameter of reconfiguration graphs for vertex colourings

Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, Daniël Paulusma
2011 Electronic Notes in Discrete Mathematics
Theorem 3.4 The class of chordal bipartite graphs is 2-colour-dense.  ...  bipartite graph we can always find two vertices u and v with N (u) ⊆ N (v). Consequently, we can define S = N (u) in order to satisfy conditions (i) and (ii). We omit the details.  ...  Because chordal bipartite graphs are bipartite, they are 2-colourable.  ...

### Switch-based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs [article]

Pieter Kleer, Viresh Patel, Fabian Stroh
2020 arXiv   pre-print
We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on n vertices.  ...  As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least n/2+7 can be transformed into each other by switch operations of size at most 10, implying that  ...  A Rapid mixing of switch-based chains for sampling 2-factors in dense graphs This section is a modification of certain parts in [1] .  ...

### Extraction and classification of dense communities in the web

Yon Dourisboure, Filippo Geraci, Marco Pellegrini
2007 Proceedings of the 16th international conference on World Wide Web - WWW '07
The core of our contribution is a new scalable algorithm for finding relatively dense subgraphs in massive graphs.  ...  More formally communities have been associated in the computer science literature with the existence of a locally dense sub-graph of the web-graph (where web pages are nodes and hyper-links are arcs of  ...  WebBase graph by a factor 0.17 in node count and 0.059 in the Edge count.  ...

### Bipartite independence number in graphs with bounded maximum degree [article]

Maria Axenovich and Jean-Sébastien Sereni and Richard Snyder and Lea Weber
2020 arXiv   pre-print
We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size t in a bipartite graph G is a copy of K_t, t in the bipartite complement of G.  ...  Let f(n, Δ) be the largest k for which every n × n bipartite graph with maximum degree Δ in one of the parts has a bi-hole of size k.  ...  Lemma 3.1 can then be applied to the bipartite graph induced on parts (T, B \ S), as in this bipartite graph every vertex in T has degree at most 2.  ...

### A local algorithm for finding dense subgraphs

Reid Andersen
2010 ACM Transactions on Algorithms
In this paper, we present a local algorithm for finding dense subgraphs of bipartite graphs, according to the measure of density proposed by Kannan and Vinay.  ...  The running time of the algorithm is O(∆k 2 ), independent of the number of vertices in the graph.  ...  We remark that any algorithm for finding dense subgraphs in undirected bipartite graphs can also be applied to find dense subgraphs in arbitary directed graphs, by a simple reduction descibed in Section  ...
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