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Rapport (Rapport De Recherche) Année : 2019

Decomposing degenerate graphs into locally irregular subgraphs

Résumé

A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition E1,...,Ek of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in the recent years. This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable k-degenerate graphs decompose into at most 3k+1 locally irregular subgraphs, which improves on previous results whenever k≤9. We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, k-trees, and planar graphs.
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Dates et versions

hal-02090804 , version 1 (05-04-2019)
hal-02090804 , version 2 (11-04-2019)
hal-02090804 , version 3 (10-04-2020)

Identifiants

  • HAL Id : hal-02090804 , version 1

Citer

Julien Bensmail, François Dross, Nicolas Nisse. Decomposing degenerate graphs into locally irregular subgraphs. [Research Report] Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900 Sophia Antipolis, France. 2019. ⟨hal-02090804v1⟩
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