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Star arboricity. (English) Zbl 0780.05043
C. St. J. A. Nash-Williams [J. Lond. Math. Soc. 39, 12 (1964; Zbl 0119.388)] defined the arboricity of a graph $$G$$, shortly $$A(G)$$, as the minimum number of forests needed to cover all edges of $$G$$. By a star forest the authors mean a forest all of whose components are stars. J. Akiyama and M. Kano introduced the star arboricity of $$G$$, denoted $$\text{st}(G)$$, as follows: $$\text{st}(G)$$ is the minimum number of star forests whose union covers all edges of $$G$$. Let $$\Delta$$ be the minimum degree of a vertex in $$G$$. In the paper under review it is shown that for any graph $$G$$, $$\text{st}(G)\leq A(G)+O(\log \Delta)$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 60C05 Combinatorial probability
##### Keywords:
arboricity; star forest; star arboricity
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##### References:
 [1] J. Akiyama andM. Kano: Path factors of a graph, in:Graph Theory and its Applications, Wiley and Sons, New York, 1984. [2] Ilan Algor andNoga Alon: The star arboricity of graphs,Discrete Math.,75 (1989), 11-22. · Zbl 0684.05033 · doi:10.1016/0012-365X(89)90073-3 [3] Y. Aoki: The star arboricity of the complete regular multipartite graphs, preprint. · Zbl 0737.05038 [4] P. Erd?s andL. Lovász: Problems and results on 3-chromatic hypergraphs and some related question, in:Infinite and Finite Sets, A. Hajnal et al. editors, North Holland, Amsterdam, 1975, 609-628. [5] C. St. J. A. Nash-Williams: Decomposition of finite graphs into forests,J. London Math. Soc. 39 (1964), 12. · Zbl 0119.38805 · doi:10.1112/jlms/s1-39.1.12
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