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Complete bipartite factorisations of Kn,n

Nigel Martin
2003 Discrete Mathematics  
We continue the study of complete factorisations of complete bipartite graphs of the form Kn;n by factors whose components are copies of Kp;q for given, ÿxed, p ¡ q.  ...  This paper sets out a general strategy for showing that these conditions are also su cient for p and q coprime and odd extending work of earlier papers where the case p = 1 was solved.  ...  Acknowledgements The author wishes to record his thanks to the referees for their careful reading of the manuscript and for their helpful suggestions in its revision.  ... 
doi:10.1016/s0012-365x(02)00819-1 fatcat:dddb5azlpnhfta45umsndqccvu

Complete bipartite factorisations by complete bipartite graphs

N. Martin
1997 Discrete Mathematics  
We study complete Kp,q-factorisations of Kin, n, Simple necessary conditions are found and we conjecture that these conditions are also sufficient.  ...  A general construction is given to find infinite families of factorisations proving the conjecture in many cases. The conjecture is proved for Kl,q-factorisations of Kn, n except for q = 4k + 1.  ...  The vertex (respectively edge) set of a graph G will be denoted V(G) (respectively E(G)). Km,n will denote a complete bipartite graph on sets of size m and n.  ... 
doi:10.1016/s0012-365x(97)00248-3 fatcat:onzhpz5bmjg3fj76aoqnc36aui

Page 52 of Mathematical Reviews Vol. , Issue 99a [page]

1991 Mathematical Reviews  
Here, the au- thors prove that if G is a simple connected graph and k > 2 is an integer such that (i) kn is even, (ii) n > 9k — 1 —4)/2(k — 1)?  ...  We present a complete solution of this problem for the case D > 4.  ... 

Page 3898 of Mathematical Reviews Vol. , Issue 81J [page]

1981 Mathematical Reviews  
Robinson (Newcastle) Quinn, Stephen J. 81j:05072 Factorisation of complete bipartite graphs into two isomorphic subgraphs. Combinatorial mathematics, VI (Proc. Sixth Austral. Conf., Univ.  ...  These two isomorphic subgraphs are called factors in the factorisation, and if G= Kn»  ... 

A sufficient condition for graphs to be fractional (k,m)-deleted graphs

Sizhong Zhou
2011 Applied Mathematics Letters  
This result is best possible in some sense, and it is an extension of Zhou's previous results.  ...  Let h : where H is any subgraph of G with m edges. In this paper, we use a binding number to obtain a sufficient condition for a graph to be a fractional (k, m)deleted graph.  ...  graph is bipartite).  ... 
doi:10.1016/j.aml.2011.03.041 fatcat:ewgatz7y4fb4nmkluiqud4zn6e

Hamiltonian cycles and 1-factors in 5-regular graphs [article]

Nico Van Cleemput, Carol T. Zamfirescu
2021 arXiv   pre-print
of 1-factorisations each containing at least one perfect pair.  ...  ., 10 }, there exist infinitely many 5-regular graphs of genus g containing a 1-factorisation with exactly k pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle.  ...  Zamfirescu's research is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO).  ... 
arXiv:2008.03173v2 fatcat:5cyupj3qcreenc7y6hbss4nowe

Tree amalgamations and quasi-isometries [article]

Matthias Hamann
2018 arXiv   pre-print
of their terminal factorisations.  ...  In the end, we discuss the impact of our results to a question of Woess.  ...  . * G 1 n * . . . * G kn n .  ... 
arXiv:1812.04987v1 fatcat:aqrbueoasvha5jf22tl2mafqam

Entanglement content of quantum particle excitations. III. Graph partition functions

Olalla A. Castro-Alvaredo, Cecilia De Fazio, Benjamin Doyon, István M. Szécsényi
2019 Journal of Mathematical Physics  
We consider two measures of entanglement, the logarithmic negativity and the entanglement entropy, between regions of space in excited states of many-body systems formed by a finite number of particle  ...  is given by the entanglement of natural qubit states, representing the uniform distribution of particles in space.  ...  Consider the set G k,n of all graphs as follows. They are formed by the 2kn vertices V = {V j,m : j ∈ I k , m ∈ I n , ∈ {l, r}} -two sets, "left" and "right", of kn vertices -and kn edges.  ... 
doi:10.1063/1.5098892 fatcat:7fgpcgdo35eevg3e7uwpf4tlxq

Regular embeddings of Kn,n where n is an odd prime power

Gareth A. Jones, Roman Nedela, Martin Škoviera
2007 European journal of combinatorics (Print)  
The method involves groups which factorise as a product of two cyclic groups of order n.  ...  We show that if n = p e where p is an odd prime and e ≥ 1, then the complete bipartite graph K n,n has p e−1 regular embeddings in orientable surfaces.  ...  Our aim is to consider the regular orientable embeddings of the complete bipartite graphs K n,n .  ... 
doi:10.1016/j.ejc.2005.07.021 fatcat:c25apmk6czhmvfhddekza6j2be

Master index of volumes 181–190

1998 Discrete Mathematics  
Lindner, The Doyen-Wilson theorem for maximum packings of Kn with 4-cycles Fu, H.-L., see B.-L.  ...  Halperin, Bipartite subgraphs of integer weighted graphs 181 (1998) 19-29 Alsardary, S.Y. and J. Wojciechowski, The basis number of the powers of the complete graph 188 (1998) 13-25 Amar, D. and O.  ... 
doi:10.1016/s0012-365x(98)90328-4 fatcat:s2tsivncvfcilf6jlbl4fx24zq

Norms of idempotent Schur multipliers [article]

Rupert H. Levene
2014 arXiv   pre-print
Many of our arguments are framed in the combinatorial language of bipartite graphs.  ...  We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n x n matrix, or of an n x (n+1) matrix, both have norm 2/(n+1) cot(pi/(n+1)), and we consider the average norm  ...  ,n,1/2 G q ≥ Kn | 1 q − 1 2 | .  ... 
arXiv:1302.4849v2 fatcat:e2gmzgduezgtzfi67nqyemkyju

Multicolour Turán problems

Peter Keevash, Mike Saks, Benny Sudakov, Jacques Verstraëte
2004 Advances in Applied Mathematics  
A simple k-colouring of a multigraph G is a decomposition of the edge multiset as the sum of k simple graphs, called 'colours'.  ...  For complete bipartite graphs K r,s with s > (r − 1)!  ...  The problem is more challenging for bipartite graphs that contain cycles. In this section we will be concerned with the case of complete bipartite graphs K r,s .  ... 
doi:10.1016/j.aam.2003.08.005 fatcat:umtvbqkdm5ga3cxpnf2ic3hrsq

Telecloning of qudits via partially entangled states

Gabriel Araneda, Nataly Cisternas, Aldo Delgado
2016 Quantum Information Processing  
The results presented here are valid for arbitrary numbers of copies of a single input qudit state of any dimension.  ...  We study the process of quantum telecloning of d-dimensional pure quantum states using partially entangled pure states as quantum channel.  ...  (22) is transformed into the state GXOR P X |ψ P X AC = 1 d d−1 n,m=0 d−1 k=0 c k ω kn |k P |m X ⊗ U −1 nm d−1 j=0 α j |φ j AC , (26) where the systems X and P are factorisable.  ... 
doi:10.1007/s11128-016-1348-2 fatcat:kztwd3vobjbuhkdwr63gih4suy

A survey on additive and multiplicative decompositions of sumsets and of shifted sets [chapter]

Christian Elsholtz
2009 Combinatorial Number Theory and Additive Group Theory  
The methods involved are of analytic nature such as the large sieve, and of combinatorial nature such as extremal graph theory.  ...  In this paper we survey results on sumsets with multiplicative properties and the question if a shifted copy of a multiplicatively defined set can again be multiplicatively defined.  ...  A complete bipartite graph K s,t corresponds to disjoint sets of primes A and B of sizes s and t such that all λa i + (1 − λ)b j are also prime.  ... 
doi:10.1007/978-3-7643-8962-8_16 fatcat:vm7nqjyckjdnpi4wctrn5k3ke4


2010 Glasgow Mathematical Journal  
Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with p ≥ (a+b−1)(a+b−2)+bn−2 https://doi.  ...  graph is bipartite).  ...  Suppose |N G (X)| > (k − 1)p + |X| + kn − 1 2k − 1 for every non-empty independent subset X of V (G), and δ(G) > (k − 1)p + 2k + kn − 2 2k − 1 , then G is a fractional (k, n)-critical graph.  ... 
doi:10.1017/s001708951000011x fatcat:ykjxrvd2hvha3hf23vw6myrjbq
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