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Paths, stars and the number three. (English) Zbl 0857.05052

A dominating set for a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex in \(V(G)-D\) has a neighbour in \(D\). In this paper it is proved that any graph \(G\) of order \(n\) with minimum degree at least 3 has a dominating set of size at most \(3n/8\). This is the same as saying that the graph can be covered by at most \(3n/8\) stars (a star \(S\) being a vertex \(v\) with edges to all vertices in \(S-v\)). It is also shown that any connected cubic graph \(G\) of order \(n\) can be covered by \(\lceil n/9\rceil\) vertex disjoint paths. Both of these results are sharp.

MSC:

05C35 Extremal problems in graph theory
05C38 Paths and cycles
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References:

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