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Article Dans Une Revue Discrete Mathematics Année : 2020

Partitioning sparse graphs into an independent set and a graph with bounded size components

Résumé

We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph G admits an (I, O k)-partition if its vertex set can be partitioned into an independent set and a set that induces a graph with components of order at most k. We prove that every graph G with mad(G) < 5 2 admits an (I, O 3)-partition. This implies that every planar graph with girth at least 10 can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 3. We also prove that every graph G with mad(G) < 8k 3k+1 = 8 3 1 − 1 3k+1 admits an (I, O k)-partition. This implies that every planar graph with girth at least 9 can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 9.
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Dates et versions

hal-02990588 , version 1 (16-11-2020)

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Ilkyoo Choi, François Dross, Pascal Ochem. Partitioning sparse graphs into an independent set and a graph with bounded size components. Discrete Mathematics, 2020, 343 (8), pp.111921. ⟨10.1016/j.disc.2020.111921⟩. ⟨hal-02990588⟩
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