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Bounded size components – partitions and transversals. (English) Zbl 1033.05083
The authors prove that there is an absolute constant $$C$$ such that the vertex set of every graph with maximum degree $$5$$ can be partitioned into $$2$$ parts such that the subgraph induced by each part does not have a component of size larger than $$C$$. This way they solve a problem posed by Alon at al. Moreover, they show that the vertex set of every graph with maximum degree 4 can be partitioned into 2 parts such that the subgraph induced by each part does not have a component of size larger than $$6$$. The next result says that it is possible to partition the vertex set of a graph with maximum degree at most 8 into 3 parts such that each part induces components of size at most an absolute constant $$C$$. Results concerning partitions of graphs of a given maximum degree into $$k$$ parts ($$k>2$$) are given too. Finally, the authors prove that if the vertex set of a graph $$G$$ is partitioned into subsets of size at least $$\Delta +\lfloor\Delta / r\rfloor$$ (where $$\Delta$$ is the maximum degree of $$G$$) then there exists a transversal for this partition that induces in $$G$$ a subgraph with all components bounded in size by $$r$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C35 Extremal problems in graph theory
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