Opuscula Mathematica

Opuscula Math.
 27
, no. 1
 (), 5-11
Opuscula Mathematica

Tree domatic number in graphs


Abstract. A dominating set \(S\) in a graph \(G\) is a tree dominating set of \(G\) if the subgraph induced by \(S\) is a tree. The tree domatic number of \(G\) is the maximum number of pairwise disjoint tree dominating sets in \(V(G)\). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most \(4\) and give a characterization of planar graphs with the tree domatic number \(3\).
Keywords: tree domatic number, regular graph, planar graph, Cartesian product.
Mathematics Subject Classification: 05C69, 05C35.
Cite this article as:
Xue-gang Chen, Tree domatic number in graphs, Opuscula Math. 27, no. 1 (2007), 5-11
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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