A lower bound for the packing chromatic number of the Cartesian product of cycles

Yolandé Jacobs; Elizabeth Jonck; Ernst Joubert

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1344-1357
  • ISSN: 2391-5455

Abstract

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Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ⊆ V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = {X 1, X 2, …, X k} of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q). We will also show that if t is a multiple of four, then χρ(C 4 □ C q) = 9.

How to cite

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Yolandé Jacobs, Elizabeth Jonck, and Ernst Joubert. "A lower bound for the packing chromatic number of the Cartesian product of cycles." Open Mathematics 11.7 (2013): 1344-1357. <http://eudml.org/doc/269036>.

@article{YolandéJacobs2013,
abstract = {Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ⊆ V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = \{X 1, X 2, …, X k\} of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q). We will also show that if t is a multiple of four, then χρ(C 4 □ C q) = 9.},
author = {Yolandé Jacobs, Elizabeth Jonck, Ernst Joubert},
journal = {Open Mathematics},
keywords = {Graph; Packing; Chromatic; Cartesian product; Cycles; Bounds; packing; chromatic; cycles; bounds},
language = {eng},
number = {7},
pages = {1344-1357},
title = {A lower bound for the packing chromatic number of the Cartesian product of cycles},
url = {http://eudml.org/doc/269036},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Yolandé Jacobs
AU - Elizabeth Jonck
AU - Ernst Joubert
TI - A lower bound for the packing chromatic number of the Cartesian product of cycles
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1344
EP - 1357
AB - Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ⊆ V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = {X 1, X 2, …, X k} of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q). We will also show that if t is a multiple of four, then χρ(C 4 □ C q) = 9.
LA - eng
KW - Graph; Packing; Chromatic; Cartesian product; Cycles; Bounds; packing; chromatic; cycles; bounds
UR - http://eudml.org/doc/269036
ER -

References

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  6. [6] Finbow A.S., Rall D.F., On the packing chromatic number of some lattices, Discrete Appl. Math., 2010, 158(12), 1224–1228 http://dx.doi.org/10.1016/j.dam.2009.06.001[WoS][Crossref] Zbl1221.05137
  7. [7] Goddard W., Hedetniemi S.M., Hedetniemi S.T., Harris J.M., Rall D.F., Broadcast chromatic numbers of graphs, Ars Combin., 2008, 86, 33–49 Zbl1224.05172
  8. [8] Sloper C., Broadcast-coloring in trees, Reports in Informatics, #233, University of Bergen, Bergen, 2002, available at http://www.ii.uib.no/~sloper/Papers/brep.pdf 
  9. [9] Soukal R., Holub P., A note on packing chromatic number of the square lattice, Electron. J. Combin., 2010, 17, #17 Zbl1215.05050

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