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Article Dans Une Revue Discrete Mathematics Année : 2010

Coloring the square of the Cartesian product of two cycles

Résumé

The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ in such a way that distinct vertices with distance at most two in $G$ are joined by an edge. We study the chromatic number of the square of the Cartesian product $C_m\Box C_n$ of two cycles and show that the value of this parameter is at most 7 except when $m=n=3$, in which case the value is 9, and when $m=n=4$ or $m=3$ and $n=5$, in which case the value is 8. Moreover, we conjecture that whenever $G=C_m\Box C_n$, the chromatic number of $G^2$ equals $\lceil mn/\alpha(G^2) \rceil$, where $\alpha(G^2)$ denotes the size of a maximal independent set in $G^2$.
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Dates et versions

hal-00392145 , version 1 (05-06-2009)
hal-00392145 , version 2 (18-06-2009)
hal-00392145 , version 3 (10-05-2010)

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Eric Sopena, Jiaojiao Wu. Coloring the square of the Cartesian product of two cycles. Discrete Mathematics, 2010, 310 (17-18), pp.2327-2333. ⟨10.1016/j.disc.2010.05.011⟩. ⟨hal-00392145v3⟩

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