On decomposing regular graphs into locally irregular subgraphs

Baudon, Olivier; Bensmail, Julien; Przybyło, Jakub; Woźniak, Mariusz

doi:10.1016/j.ejc.2015.02.031

Cited by 46 publications

(94 citation statements)

References 15 publications

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“…One way for dealing with this question is to consider the following edge-colouring notion. An (improper) edge-colouring φ of G is locally irregular if every colour class of φ induces a locally irregular subgraph of G. As pointed out in [4], a locally irregular 2-edge-colouring of a regular graph G is also a neighbour-sum-distinguishing 2-edge-weighting of G. Hence, studying locally irregular 2-edge-colourings of graphs can be a way to tackle the 1-2-3 Conjecture in the context of regular graphs.…”

Section: Introductionmentioning

confidence: 96%

On the complexity of determining the irregular chromatic index of a graph

Baudon¹,

Bensmail²,

Sopena³

2015

Journal of Discrete Algorithms

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An undirected simple graph G is locally irregular if adjacent vertices of G have different degrees. An edge-colouring φ of G is locally irregular if each colour class of φ induces a locally irregular subgraph of G. The irregular chromatic index χ irr (G) of G is the least number of colours used by a locally irregular edge-colouring of G (if any). We show that the problem of determining the irregular chromatic index of a graph can be handled in linear time when restricted to trees, but it remains NP-complete in general.

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Section: Introductionmentioning

confidence: 96%

On the complexity of determining the irregular chromatic index of a graph

Baudon¹,

Bensmail²,

Sopena³

2015

Journal of Discrete Algorithms

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“…whose irregular chromatic index is finite, can be decomposed into at most three locally irregular graphs [1]. This conjecture was verified for several classes of graphs, including trees, complete graphs, and regular graphs with large degree which are, in some sense, the least locally irregular graphs.…”

Section: Introductionmentioning

confidence: 87%

“…In connexion with the investigations from [1], which inspired the present work, we here consider graphs that are irregular locally, rather than totally. In particular, by a locally irregular graph, we refer to a graph whose every two adjacent vertices have distinct degrees.…”

Section: Introductionmentioning

confidence: 99%

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

Bensmail

Renault

2016

Graphs and Combinatorics

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An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E1 ∪ E2 ∪ ... ∪ E k of the edge set E(G) such that each Ei induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.

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“…The notion of locally irregular edge colouring of graphs was recently introduced by Baudon et al [3] and combines two popular notions of graph theory, namely the ones of locally irregular graphs and adjacent vertex distinguishing edge colourings of graphs. Locally irregular graphs were first introduced under the name of highly irregular graphs in a work aiming at defining some possible ways for catching the irregularity of graphs [2].…”

Section: Introductionmentioning

confidence: 99%

“…Such graphs are said non-colourable (with respect to locally irregular edge colourings). As shown in [3], non-colourable graphs include odd length paths and cycles, and a family of tree-like graphs with maximum degree at most 3 obtained by connecting an arbitrary number of triangles in a specific way. It is worth mentioning that, because of their simple structure, non-colourable graphs may be recognized in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Complexity of determining the irregular chromatic index of a graph

Bensmail

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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A graph G is locally irregular if adjacent vertices of G have different degrees. A k-edge colouring φ of G is locally irregular if each of the k colours of φ induces a locally irregular subgraph of G. The irregular chromatic index χ irr (G) of G is the least number of colours used by a locally irregular edge colouring of G (if any). We show that determining whether χ irr (G) = 2 is NP-complete, even when G is assumed to be a planar graph with maximum degree at most 6.

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On decomposing regular graphs into locally irregular subgraphs

Cited by 46 publications

References 15 publications

On the complexity of determining the irregular chromatic index of a graph

On the complexity of determining the irregular chromatic index of a graph

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

Complexity of determining the irregular chromatic index of a graph

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