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Discussiones Mathematicae Graph Theory 24(1) (2004) 73-79
DOI: https://doi.org/10.7151/dmgt.1214

ON THE HETEROCHROMATIC NUMBER
OF CIRCULANT DIGRAPHS

Hortensia Galeana-Sánchez and Víctor Neumann-Lara

Instituto de Matemáticas, UNAM
Universidad Nacional Autónoma de México
Ciudad Universitaria
04510, México, D.F., MÉXICO

Abstract

The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes.

For any two integers s and n with 1 ≤ s ≤ n, let D_n,s be the oriented graph such that V(D_n,s) is the set of integers mod 2n+1 and A(D_n,s) = {(i,j) : j− i ∈ {1,2,...,n}∖{s}}.

In this paper we prove that hc(D_n,s) ≤ 5 for n ≥ 7. The bound is tight since equality holds when s ∈ {n,[(2n+1)/3]}.

Keywords: circulant tournament, vertex colouring, heterochromatic number, heterochromatic triangle.

2000 Mathematic Subject Classification: 05C20, 05C15.

References

[1]	C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).
[2]	B. Abrego, J.L. Arocha, S. Fernández Merchant and V. Neumann-Lara, Tightness problems in the plane, Discrete Math. 194 (1999) 1-11, doi: 10.1016/S0012-365X(98)00031-4.
[3]	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.
[4]	P. Erdős, M. Simonovits and V.T. Sós, Anti-Ramsey Theorems (in: Infinite and Finite Sets, Keszthely, Hungary, 1973), Colloquia Mathematica Societatis János Bolyai 10 633-643.
[5]	H. Galeana-Sánchez and V. Neumann-Lara, A class of tight circulant tournaments, Discuss. Math. Graph Theory 20 (2000) 109-128, doi: 10.7151/dmgt.1111.
[6]	Y. Manoussakis, M. Spyratos, Zs. Tuza, M. Voigt, Minimal colorings for properly colored subgraphs, Graphs and Combinatorics 12 (1996) 345-360, doi: 10.1007/BF01858468.
[7]	V. Neumann-Lara, The acyclic disconnection of a digraph, Discrete Math. 197-198 (1999) 617-632.

Received 29 June 2001
Revised 13 May 2003