# On the heterochromatic number of circulant digraphs

• Volume: 24, Issue: 1, page 73-79
• ISSN: 2083-5892

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## Abstract

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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 $V\left({D}_{n,s}\right)$ is the set of integers mod 2n+1 and $A\left({D}_{n,s}\right)=\left(i,j\right):j-i\in 1,2,...,n\setminus s..$In this paper we prove that $hc\left({D}_{n,s}\right)\le 5$ for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].

## How to cite

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Hortensia Galeana-Sánchez, and Víctor Neumann-Lara. "On the heterochromatic number of circulant digraphs." Discussiones Mathematicae Graph Theory 24.1 (2004): 73-79. <http://eudml.org/doc/270660>.

@article{HortensiaGaleana2004,
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].},
author = {Hortensia Galeana-Sánchez, Víctor Neumann-Lara},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {circulant tournament; vertex colouring; heterochromatic number; heterochromatic triangle},
language = {eng},
number = {1},
pages = {73-79},
title = {On the heterochromatic number of circulant digraphs},
url = {http://eudml.org/doc/270660},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - Víctor Neumann-Lara
TI - On the heterochromatic number of circulant digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2004
VL - 24
IS - 1
SP - 73
EP - 79
AB - 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].
LA - eng
KW - circulant tournament; vertex colouring; heterochromatic number; heterochromatic triangle
UR - http://eudml.org/doc/270660
ER -

## References

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1. [1] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).
2. [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. Zbl0931.05030
3. [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. Zbl0776.05079
4. [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. [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. Zbl0969.05031
6. [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. Zbl0864.05035
7. [7] V. Neumann-Lara, The acyclic disconnection of a digraph, Discrete Math. 197-198 (1999) 617-632. Zbl0928.05033

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