Geodesic
Connected domatic number in planar graphs
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 173-179 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The connected domatic number of $G$ is the maximum number of pairwise disjoint, connected dominating sets in $V(G)$. We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.
A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The connected domatic number of $G$ is the maximum number of pairwise disjoint, connected dominating sets in $V(G)$. We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.
Classification : 05C10, 05C40, 05C69, 05C70, 05C99
Keywords: connected dominating set; connected domatic number; planar 
@article{CMJ_2001_51_1_a15,
     author = {Hartnell, Bert L. and Rall, Douglas F.},
     title = {Connected domatic number in planar graphs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {173--179},
     year = {2001},
     volume = {51},
     number = {1},
     mrnumber = {1814642},
     zbl = {1079.05512},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a15/}
}
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Hartnell, Bert L.; Rall, Douglas F. Connected domatic number in planar graphs. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 173-179. http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a15/