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Complete bipartite factorisations of Kn,n

2003
*
Discrete Mathematics
*

We continue the study

doi:10.1016/s0012-365x(02)00819-1
fatcat:dddb5azlpnhfta45umsndqccvu
*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. ...##
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Complete bipartite factorisations by complete bipartite graphs

1997
*
Discrete Mathematics
*

We study

doi:10.1016/s0012-365x(97)00248-3
fatcat:onzhpz5bmjg3fj76aoqnc36aui
*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*. ...##
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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. ...##
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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*» ...##
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A sufficient condition for graphs to be fractional (k,m)-deleted graphs

2011
*
Applied Mathematics Letters
*

This result is best possible in some sense, and it is an extension

doi:10.1016/j.aml.2011.03.041
fatcat:ewgatz7y4fb4nmkluiqud4zn6e
*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*). ...##
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Hamiltonian cycles and 1-factors in 5-regular graphs
[article]

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). ...

##
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Tree amalgamations and quasi-isometries
[article]

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*. ...

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

2019
*
Journal of Mathematical Physics
*

We consider two measures

doi:10.1063/1.5098892
fatcat:7fgpcgdo35eevg3e7uwpf4tlxq
*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. ...##
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Regular embeddings of Kn,n where n is an odd prime power

2007
*
European journal of combinatorics (Print)
*

The method involves groups which

doi:10.1016/j.ejc.2005.07.021
fatcat:c25apmk6czhmvfhddekza6j2be
*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*. ...##
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Master index of volumes 181–190

1998
*
Discrete Mathematics
*

Lindner, The Doyen-Wilson theorem for maximum packings

doi:10.1016/s0012-365x(98)90328-4
fatcat:s2tsivncvfcilf6jlbl4fx24zq
*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. ...##
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Norms of idempotent Schur multipliers
[article]

2014
*
arXiv
*
pre-print

Many

arXiv:1302.4849v2
fatcat:e2gmzgduezgtzfi67nqyemkyju
*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 | . ...##
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Multicolour Turán problems

2004
*
Advances in Applied Mathematics
*

A simple k-colouring

doi:10.1016/j.aam.2003.08.005
fatcat:umtvbqkdm5ga3cxpnf2ic3hrsq
*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 . ...##
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Telecloning of qudits via partially entangled states

2016
*
Quantum Information Processing
*

The results presented here are valid for arbitrary numbers

doi:10.1007/s11128-016-1348-2
fatcat:kztwd3vobjbuhkdwr63gih4suy
*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*. ...##
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A survey on additive and multiplicative decompositions of sumsets and of shifted sets
[chapter]

2009
*
Combinatorial Number Theory and Additive Group Theory
*

The methods involved are

doi:10.1007/978-3-7643-8962-8_16
fatcat:vm7nqjyckjdnpi4wctrn5k3ke4
*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. ...##
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A SUFFICIENT CONDITION FOR A GRAPH TO BE A FRACTIONAL ( f, n)-CRITICAL GRAPH

2010
*
Glasgow Mathematical Journal
*

Let a, b and

doi:10.1017/s001708951000011x
fatcat:ykjxrvd2hvha3hf23vw6myrjbq
*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://www.cambridge.org/core/terms. 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. ...
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