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The square chromatic number of the torus

Goodarzvand Chegini, A ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2015.09.003
  3. Publisher: Elsevier
  4. Abstract:
  5. The square of a graph G denoted by G2, is the graph with the same vertex set as G and edges linking pairs of vertices at distance at most 2 in G. The chromatic number of the square of the Cartesian product of two cycles was previously determined for some cases. In this paper, we determine the precise value of χ((Cm□Cn)2) for all the remaining cases. We show that for all ordered pairs (m,n) except for (7,11) we have χ(Cm□Cn)2)=γV((Cm□Cn)2)|α((Cm□Cn)2), where α(G) denotes the independent number of G. This settles a conjecture of Sopena and Wu (2010). We also show that the smallest integer k such that χ(Cm□Cn2)≤6 for every m,≥k is 10. This answers a question of Shao and Vesel (2013)
  6. Keywords:
  7. 2-distance colouring ; Cartesian product ; Torus
  8. Source: Discrete Mathematics ; Volume 339, Issue 2 , 2016 , Pages 447-456 ; 0012365X (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0012365X15003131