Decomposing degenerate graphs into locally irregular subgraphs

Julien Bensmail 1 François Dross 1 Nicolas Nisse 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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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https://hal.archives-ouvertes.fr/hal-02090804
Contributor : Julien Bensmail <>
Submitted on : Thursday, April 11, 2019 - 5:55:58 PM
Last modification on : Wednesday, April 17, 2019 - 1:20:36 AM

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Julien Bensmail, François Dross, Nicolas Nisse. Decomposing degenerate graphs into locally irregular subgraphs. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France. 2019. ⟨hal-02090804v2⟩

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