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P3-Factorization of complete bipartite graphs

1988
*
Discrete Mathematics
*

In this paper, it is shown that a necessary and sufficient condition for the existence

doi:10.1016/0012-365x(88)90227-0
fatcat:gzzklvbl45ejrnt2fe72v4rdfi
*of*a P,-*factorization**of*K"" is (i) m + n -0 (mod 3), (ii) m < 2n, (iii) n s 2m and (iv) 3mn/2(m + n) is an integer ... Introduction Let*P3*be a path on 3 points and K,,,, be a*complete**bipartite**graph*with partite sets VI and V,, where IV,1 = m and IV,/ = n. ... In this*graph*, two points are adjacent if and only if the subsets come from disjoint independent sets*of*K,,,,,. G is a*complete**bipartite**graph*KS,,. ...##
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Imbeddings of the tensor product of graphs where the second factor is a complete graph

1998
*
Discrete Mathematics
*

In this paper we present genus result for the tensor product

doi:10.1016/s0012-365x(97)00130-1
fatcat:qqjsgziwo5eyblhlwtkhpoxvza
*of**graphs*where the second*factor*is the*complete**graph*Km and m is a power*of*2. ... The genus*of*H ® K." where H is a*graph*with a quadrilateral imbedding and also H is either,*bipartite*, with bichromatic dual, or with straight-ahead imbedding is given * ... In this paper we will use the surgical-voltage method to find the genus parameter for the tensor product*of**graphs*when one*of*the*factors*is Kin, the*complete**graph*on m vertices where m is a power*of*...##
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Perfect weak modular product graphs
[article]

2018
*
arXiv
*
pre-print

Interesting cases include

arXiv:1809.09939v1
fatcat:25tp4y5xlveerdsowfhp7geu4u
*complete*multipartite*graphs*and disjoint unions*of*cliques. ... The weak modular product differs from the direct product by also encoding non-adjacencies*of*the*factor**graphs*in its edges. ... The software SageMath [Sage] has been invaluable to this work, alongside the ISGCI*graph*database [ISGCI]. ...##
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Page 1724 of Mathematical Reviews Vol. , Issue 2004c
[page]

2004
*
Mathematical Reviews
*

. the existence

*of*a*P3*-*factorization**of*the symmetric*complete** myn? ... Summary: “In this paper, a necessary and sufficient condition for the existence*of*a*P3*-*factorization**of*the*complete**bipartite*multigraph AK,,,, and a necessary and sufficient condition for . ——- . . ...##
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Packing paths of length at least two

2004
*
Discrete Mathematics
*

vertices

doi:10.1016/j.disc.2004.01.016
fatcat:gena2ana7zhzbhrijfbj34hkhe
*of*the original*graph*. ... We give a simple proof for Kaneko's theorem which gives a su cient and necessary condition for the existence*of*vertex disjoint paths in a*graph*, each*of*length at least two, that altogether cover all ... result*of*Kaneko and started to work on this paper. ...##
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Packing paths of length at least two

2004
*
Discrete Mathematics
*

vertices

doi:10.1016/s0012-365x(04)00152-9
fatcat:sdx5yermf5gutcp5ljibjsnofm
*of*the original*graph*. ... We give a simple proof for Kaneko's theorem which gives a su cient and necessary condition for the existence*of*vertex disjoint paths in a*graph*, each*of*length at least two, that altogether cover all ... result*of*Kaneko and started to work on this paper. ...##
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Page 8208 of Mathematical Reviews Vol. , Issue 99m
[page]

1999
*
Mathematical Reviews
*

Gary MacGillivray (3-VCTR; Victoria, BC)
99m:05119 05C70
Du, Beiliang (PRC-SOO; Suzhou)

*P3*-*factorization**of**complete*multipartite*graphs*. ... Summary: “In this note it is shown that a necessary and sufficient condition for the existence*of*a*P3*-*factorization**of*the*complete*multipartite*graph*AK” is (1) m > 3, (2) mn =0 (mod 3) and (3) m A(m ...##
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Expression of Composite Number as Primes Using Hyper Graphs

2015
*
International Journal of Soft Computing, Mathematics and Control
*

KEY WORDS prime

doi:10.14810/ijscmc.2015.4101
fatcat:bqdgxooeozhcnlwiuul2bwqf3q
*factorization*theorem,*complete*prime vertex composite edge weighted hypergraph ... The prime*factorization*theorem states that every integer greater than 1 can be expressed as a product*of*primes. ... The prime*factorization*is not unique in case*of*connected*bipartite**graphs*. They have proved that prime*factorization**of*a connected*bipartite**graph*has exactly one*bipartite**factor*. ...##
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On another Boolean matrix

1980
*
Theoretical Computer Science
*

In his paper "On a Boolean matrix", Nechiporuk gave an explicit example

doi:10.1016/0304-3975(80)90034-1
fatcat:tuynnafxdbfovdhnrjuyg6sr7u
*of*a set*of*n homogeneous monotone Boolean functions*of*the first degree in n variables that require f2(n3'*) two-input gates in ... The conditions we desire for Boolean matrices can then be translated into conditions on*bipartite**graphs*; specifically, we seek*bipartite**graphs*that have many edges but no large*complete**bipartite*subgraphs ... Nechiporuk's results then follow from a construction (due to Sos and Turin [3] )*of*a*bipartite**graph*with n(n"') edges but no copy*of*K2,* (that is, no*complete**bipartite*subgraph with 2 inputs and 2 ...*of*18

*Graph*representation I The rating matrix describes a

*bipartite*

*graph*. ...

*Bipartite*Aware Greedy Algorithm I Real word

*bipartite*

*graphs*are often significantly skewed: one

*of*the two sets is much bigger than the other. ... Communication cost I T = the training set I U = users set I I = items set I V = U [ I I C = processing nodes I RF = replication

*factor*I RF U = users' RF

*of*18 Conclusions I three distinct contributions ...

##
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A Comparison on the Bounds of Chromatic Preserving Number and Dom-Chromatic Number of Cartesian Product and Kronecker Product of Paths

2012
*
Mapana Journal of Sciences
*

The Kronecker product G1 Ù G2

doi:10.12723/mjs.23.3
fatcat:2qxstasa45fhzdggrvuu3jabwu
*of*two*graphs*G1 = (V1, E1) and G2 = (V2, E2) is the*graph*G with vertex set V1 x V2 and any two distinct vertices (u1, v1) and (u2, v2)*of*G are adjacent if u1u2 Î E1 and ... The minimum cardinality*of*a dom-chromatic set in a*graph*G is called the dom-chromatic number (or dc- number)*of*G and is denoted by γch(G). ... The*complete**graph*K2 is an example*of*a 2-critical*graph*and odd cycles are 3-critical*graphs*. ...##
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Page 7839 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

Kaneko gave a necessary and sufficient condition for a

*graph*to have a {*P3*, Ps, Ps}-*factor*. As a corollary, he proved that every cubic*graph*has a {*P3*, Py, Ps}-*factor*. ... (I-BASI; Potenza) On determinants and permanents*of*minimally 1-*factorable*cubic*bipartite**graphs*. (English summary) Note Mat. 20 (2000/01), no. 1, 37—42. ...##
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On the genus of the tensor product of graphs where one factor is a regular graph

1994
*
Discrete Mathematics
*

This construction can also be extended to the tensor product HOG where G is a (2k+ 1)-regular

doi:10.1016/0012-365x(93)e0057-b
fatcat:7ehxteqeanayxex6q5jinfa3am
*graph*with l-*factor*. ... Assuming that H is suitably imbedded in some orientable surface by modifying the edges*of*H according to the configuration*of*G we get the permutation voltage*graph*H c2') whose permutation derived*graph*... Here we try to extend this construction to the tensor product*of**graphs*where one*of*the*factors*is either 2k-regular or (2k+ 1)-regular*graph*with a l-*factor*. ...##
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Quasirandom Graphs and the Pantograph Equation
[article]

2021
*
arXiv
*
pre-print

This provides a natural example

arXiv:2101.08173v2
fatcat:7zbcytkbnvc3td7rk2wrd4o56y
*of*an infinite family*of**graphs*, which is not forcing, and answers a natural question posed by P. Horn. ... In this article we describe a new surprising application*of*these objects in*graph*theory, by showing that the set*of*all cliques is not forcing for quasirandomness. ... If G is a*complete**bipartite**graph*1 with both parts*of*size n/2, where n is a large even number, and H is a star with k edges, i.e., a*complete**bipartite**graph*with part sizes 1 and k, then G contains ...##
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Page 623 of Mathematical Reviews Vol. , Issue 87b
[page]

1987
*
Mathematical Reviews
*

A

*bipartite*directed*graph*D is said to be*complete**bipartite*if each pair*of*vertices from different*bipartition*classes*of*D is joined by at least one edge. ... A directed*graph*is said to be traceable if it has a directed Hamiltonian path. The main theorem states that a*complete**bipartite*directed*graph*D is traceable if and only if it has an almost-*factor*. ...
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