DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 23(1) (2003) 177-187

THE SIZE OF MINIMUM 3-TREES: CASES 0 AND 1 MOD 12

Jorge L. Arocha

Instituto de Matemáticas, UNAM
Ciudad Universitaria, Circuito exterior
México 04510

e-mail:
arocha@math.unam.mx

Joaquín Tey

Departamento de Matemáticas, UAM-Iztapalapa
Ave. Sn. Rafael Atlixco #186, Col. Vicentina
México 09340
e-mail: jtey@xanum.uam.mx

Abstract

A 3-uniform hypergraph is called a minimum 3-tree, if for any 3-coloring of its vertex set there is a heterochromatic triple and the hypergraph has the minimum possible number of triples. There is a conjecture that the number of triples in such 3-tree is ⎡[(n(n−2))/3]⎤ for any number of vertices n. Here we give a proof of this conjecture for any n ≡ 0,1 mod 12.

Keywords: tight hypergraphs, triple systems.

2000 Mathematics Subject Classification: 05B07, 05C65, 05D10.

References

[1] J.L. Arocha, J. Bracho and V. Neumann-Lara, On the minimum size of tight hypergraphs, J. Graph Theory 16 (1992) 319-326, doi: 10.1002/jgt.3190160405.
[2] J.L. Arocha and J. Tey, The size of minimum 3-trees: Cases 3 and 4 mod 6, J. Graph Theory 30 (1999) 157-166, doi: 10.1002/(SICI)1097-0118(199903)30:3<157::AID-JGT1>3.0.CO;2-S.
[3] J.L. Arocha and J. Tey, The size of minimum 3-trees: Case 2 mod 3, Bol. Soc. Mat. Mexicana (3) 8 no. 1 (2002) 1-4.
[4] L. Lovász, Topological and algebraic methods in graph theory, in: Graph Theory and Related Topics, Proceedings of Conference in Honour of W.T. Tutte, Waterloo, Ontario 1977, (Academic Press, New York, 1979) 1-14.

Received 26 November 2001
Revised 6 May 2002