Balanced Bowtie and Trefoil Decomposition of Complete Tripartite Multigraphs

Kazuhiko USHIO; Hideaki FUJIMOTO

Balanced Bowtie and Trefoil Decomposition of Complete Tripartite Multigraphs

Kazuhiko USHIO, Hideaki FUJIMOTO

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First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 6) for λ 1,5 (mod 6), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 2,4 (mod 6), (iii) n₁=n₂=n₃ 0 (mod 2) for λ 3 (mod 6), and (iv) n₁=n₂=n₃ 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 3,6 (mod 9), and (iii) n₁=n₂=n₃ 3 for λ 0 (mod 9).

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E84-A No.3 pp.839-844

Publication Date: 2001/03/01

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Type of Manuscript: PAPER

Category: Graphs and Networks

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Kazuhiko USHIO, Hideaki FUJIMOTO, "Balanced Bowtie and Trefoil Decomposition of Complete Tripartite Multigraphs" in IEICE TRANSACTIONS on Fundamentals, vol. E84-A, no. 3, pp. 839-844, March 2001, doi: .
Abstract: First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 6) for λ 1,5 (mod 6), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 2,4 (mod 6), (iii) n₁=n₂=n₃ 0 (mod 2) for λ 3 (mod 6), and (iv) n₁=n₂=n₃ 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 3,6 (mod 9), and (iii) n₁=n₂=n₃ 3 for λ 0 (mod 9).
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_3_839/_p

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@ARTICLE{e84-a_3_839,
author={Kazuhiko USHIO, Hideaki FUJIMOTO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Balanced Bowtie and Trefoil Decomposition of Complete Tripartite Multigraphs},
year={2001},
volume={E84-A},
number={3},
pages={839-844},
abstract={First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 6) for λ 1,5 (mod 6), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 2,4 (mod 6), (iii) n₁=n₂=n₃ 0 (mod 2) for λ 3 (mod 6), and (iv) n₁=n₂=n₃ 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 3,6 (mod 9), and (iii) n₁=n₂=n₃ 3 for λ 0 (mod 9).},
keywords={},
doi={},
ISSN={},
month={March},}

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TY - JOUR
TI - Balanced Bowtie and Trefoil Decomposition of Complete Tripartite Multigraphs
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 839
EP - 844
AU - Kazuhiko USHIO
AU - Hideaki FUJIMOTO
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E84-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2001
AB - First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 6) for λ 1,5 (mod 6), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 2,4 (mod 6), (iii) n₁=n₂=n₃ 0 (mod 2) for λ 3 (mod 6), and (iv) n₁=n₂=n₃ 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ K_n1,_n2,_n3 is (i) n₁=n₂=n₃ 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n₁=n₂=n₃ 0 (mod 3) for λ 3,6 (mod 9), and (iii) n₁=n₂=n₃ 3 for λ 0 (mod 9).
ER -