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On locally irregular decompositions of subcubic graphs

Baudon O., Bensmail J., Hocquard H., Senhaji M., Sopena E.

Opuscula Mathematica

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2018

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Vol. 38, no. 6

795--817

EN

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 E1,.. . , Ek of E(G) such that each G[Ei] 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, Przybyło and Woźniak 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 at most 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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Recursively arbitrarily vertex-decomposable graphs

Baudon O., Gilbert F., Woźniak M.

Opuscula Mathematica

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2012

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Vol. 32, no. 4

689-706

EN

A graph G = (V, E) is arbitrarily vertex decomposable if for any sequence ϒ of positive integers adding up to/V/, there is a sequence of vertex-disjoint subsets of V whose orders are given by ϒ, and which induce connected graphs. The main aim of this paper is to study the recursive version of this problem. We present a solution for trees, suns, and partially for a class of 2-connected graphs called balloons.

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Recursively arbitrarily vertex-decomposable suns

Baudon O., Gilbert F., Woźniak M.

Opuscula Mathematica

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2011

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Vol. 31, no. 4

533-547

EN

A graph G = (V,E) is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to /V /, there is a sequence of vertex-disjoint subsets of V whose orders are given by τ , and which induce connected graphs. The aim of this paper is to study the recursive version of this problem on a special class of graphs called suns. This paper is a complement of [O. Baudon, F. Gilbert, M. Woźniak, Recursively arbitrarily vertex-decomposable graphs, research report, 2010].