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Article Dans Une Revue Journal of Graph Theory Année : 2020

Edge‐decomposing graphs into coprime forests

Résumé

Abstract The Barát‐Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree , there is a constant such that every ‐edge‐connected graph with number of edges (size) divisible by the size of admits an edge partition into copies of (a ‐decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in Bensmail et al. (2019) that when is a path with edges, there is a constant such that every 24‐edge‐connected graph with size divisible by and minimum degree has a ‐decomposition. We show in this paper that when is a coprime forest (the sizes of its components being a coprime set of integers), any graph with sufficiently large minimum degree has an ‐decomposition provided that the size of divides the size of (no connectivity is required). A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree , any graph of size divisible by the size of with sufficiently high minimum degree has a ‐decomposition, provided that is sufficiently highly connected in terms of the maximal degree of . The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3.

Dates et versions

hal-04533479 , version 1 (04-04-2024)

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Tereza Klimošová, Stéphan Thomassé. Edge‐decomposing graphs into coprime forests. Journal of Graph Theory, 2020, 97 (1), pp.21-33. ⟨10.1002/jgt.22638⟩. ⟨hal-04533479⟩
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