On locally irregular decompositions of subcubic graphs

Olivier Baudon 1 Julien Bensmail 2 Hervé Hocquard 1 Mohammed Senhaji 1 Eric Sopena 1
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E_1, ..., E_k of E(G) such that each G[E_i] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybylo and Wozniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only 4 when the maximum average degree is less than 12/5. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs.
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[Research Report] Université de bordeaux; Université de Nice - Sophia Antipolis. 2016
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Dernière modification le : vendredi 18 novembre 2016 - 01:12:00
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Olivier Baudon, Julien Bensmail, Hervé Hocquard, Mohammed Senhaji, Eric Sopena. On locally irregular decompositions of subcubic graphs. [Research Report] Université de bordeaux; Université de Nice - Sophia Antipolis. 2016. <hal-01398228>

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