# Connected domatic number in planar graphs

Bert L. Hartnell; Douglas F. Rall

Czechoslovak Mathematical Journal (2001)

- Volume: 51, Issue: 1, page 173-179
- ISSN: 0011-4642

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topHartnell, Bert L., and Rall, Douglas F.. "Connected domatic number in planar graphs." Czechoslovak Mathematical Journal 51.1 (2001): 173-179. <http://eudml.org/doc/30624>.

@article{Hartnell2001,

abstract = {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.},

author = {Hartnell, Bert L., Rall, Douglas F.},

journal = {Czechoslovak Mathematical Journal},

keywords = {connected dominating set; connected domatic number; planar; connected dominating set; connected domatic number; planar},

language = {eng},

number = {1},

pages = {173-179},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Connected domatic number in planar graphs},

url = {http://eudml.org/doc/30624},

volume = {51},

year = {2001},

}

TY - JOUR

AU - Hartnell, Bert L.

AU - Rall, Douglas F.

TI - Connected domatic number in planar graphs

JO - Czechoslovak Mathematical Journal

PY - 2001

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 51

IS - 1

SP - 173

EP - 179

AB - 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.

LA - eng

KW - connected dominating set; connected domatic number; planar; connected dominating set; connected domatic number; planar

UR - http://eudml.org/doc/30624

ER -

## References

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- Combinatorial Problems on Chessboards: II, Chapter 6, Domination in Graphs: Advanced Topics, Marcel Dekker, Inc., New York, 1997. (1997)
- Connected domination in graphs, Graph Theory and Combinatorics, Academic Press, London-New York, 1984, pp. 209–217. (1984) MR0777177
- The connected domination number of a graph, J. Math. Phys. Sci. 13 (1979), 607–613. (1979) MR0575817
- Connected domatic number of a graph, Math. Slovaca 36 (1986), 387–392. (1986) Zbl0625.05042MR0871778

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