# Rectangular table negotiation problem revisited

Open Mathematics (2011)

• Volume: 9, Issue: 5, page 1114-1120
• ISSN: 2391-5455

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## Abstract

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We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days, every member of each delegation can communicate with every member of the other delegation exactly once. We denote by H(k, 3) the graph describing the communication at one table and by mH(k, 3) the graph consisting of m disjoint copies of H(k, 3). We completely characterize all complete bipartite graphs K n,n that can be factorized into factors isomorphic to G =mH(k, 3) for k ≡ 2 (mod 4) by showing that the necessary conditions n=mk and m ≡ 0 mod(3k−2)/4 are also sufficient. This results complement previous characterizations for k ≡ 0, 1, 3 (mod 4) to settle the problem in full.

## How to cite

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Dalibor Froncek, and Michael Kubesa. "Rectangular table negotiation problem revisited." Open Mathematics 9.5 (2011): 1114-1120. <http://eudml.org/doc/269463>.

@article{DaliborFroncek2011,
abstract = {We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days, every member of each delegation can communicate with every member of the other delegation exactly once. We denote by H(k, 3) the graph describing the communication at one table and by mH(k, 3) the graph consisting of m disjoint copies of H(k, 3). We completely characterize all complete bipartite graphs K n,n that can be factorized into factors isomorphic to G =mH(k, 3) for k ≡ 2 (mod 4) by showing that the necessary conditions n=mk and m ≡ 0 mod(3k−2)/4 are also sufficient. This results complement previous characterizations for k ≡ 0, 1, 3 (mod 4) to settle the problem in full.},
author = {Dalibor Froncek, Michael Kubesa},
journal = {Open Mathematics},
keywords = {Oberwolfach problem; Graph decomposition; Graph factorization},
language = {eng},
number = {5},
pages = {1114-1120},
title = {Rectangular table negotiation problem revisited},
url = {http://eudml.org/doc/269463},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Dalibor Froncek
AU - Michael Kubesa
TI - Rectangular table negotiation problem revisited
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 1114
EP - 1120
AB - We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days, every member of each delegation can communicate with every member of the other delegation exactly once. We denote by H(k, 3) the graph describing the communication at one table and by mH(k, 3) the graph consisting of m disjoint copies of H(k, 3). We completely characterize all complete bipartite graphs K n,n that can be factorized into factors isomorphic to G =mH(k, 3) for k ≡ 2 (mod 4) by showing that the necessary conditions n=mk and m ≡ 0 mod(3k−2)/4 are also sufficient. This results complement previous characterizations for k ≡ 0, 1, 3 (mod 4) to settle the problem in full.
LA - eng
KW - Oberwolfach problem; Graph decomposition; Graph factorization
UR - http://eudml.org/doc/269463
ER -

## References

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1. [1] Colbourn C.J., Dinitz J.H. (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press Ser. Discrete Math. Appl., CRC Press, Boca Raton, 1996 Zbl0836.00010
2. [2] Chitra V., Muthusamy A., Bipartite variation of the cheesecake factory problem (in preparation) Zbl1238.05207
3. [3] El-Zanati S.I., Vanden Eynden C., On Rosa-type labelings and cyclic graph decompositions, Math. Slovaca, 2009, 59(1), 1–18 http://dx.doi.org/10.2478/s12175-008-0108-x Zbl1199.05299
4. [4] Froncek D., Oberwolfach rectangular table negotiation problem, Discrete Math., 2009, 309(2), 501–504 http://dx.doi.org/10.1016/j.disc.2008.02.034 Zbl1160.05047
5. [5] Piotrowski W.-L., The solution of the bipartite analogue of the Oberwolfach problem, Discrete Math., 97(1–3), 1991, 339–356 http://dx.doi.org/10.1016/0012-365X(91)90449-C Zbl0765.05080
6. [6] Ringel G., Lladó A.S., Serra O., Decomposition of complete bipartite graphs into trees, DMAT Research Report, Univ. Politecnica de Catalunya, 1996, #11

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