# The size of minimum 3-trees: cases 0 and 1 mod 12

Discussiones Mathematicae Graph Theory (2003)

- Volume: 23, Issue: 1, page 177-187
- ISSN: 2083-5892

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topJorge L. Arocha, and Joaquín Tey. "The size of minimum 3-trees: cases 0 and 1 mod 12." Discussiones Mathematicae Graph Theory 23.1 (2003): 177-187. <http://eudml.org/doc/270488>.

@article{JorgeL2003,

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.},

author = {Jorge L. Arocha, Joaquín Tey},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {tight hypergraphs; triple systems; rainbow coloring; 3-uniform hypergraph},

language = {eng},

number = {1},

pages = {177-187},

title = {The size of minimum 3-trees: cases 0 and 1 mod 12},

url = {http://eudml.org/doc/270488},

volume = {23},

year = {2003},

}

TY - JOUR

AU - Jorge L. Arocha

AU - Joaquín Tey

TI - The size of minimum 3-trees: cases 0 and 1 mod 12

JO - Discussiones Mathematicae Graph Theory

PY - 2003

VL - 23

IS - 1

SP - 177

EP - 187

AB - 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.

LA - eng

KW - tight hypergraphs; triple systems; rainbow coloring; 3-uniform hypergraph

UR - http://eudml.org/doc/270488

ER -

## References

top- [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. Zbl0776.05079
- [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 Zbl0917.05053
- [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. Zbl1006.05011
- [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.

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