How to Define an Irregular Graph

Chartrand, Gary; Erdös, Paul; Oellermann, Ortrud R.

doi:10.2307/2686701

Cited by 47 publications

(44 citation statements)

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“…The similar concept of irregularity strength, where one concerns sums instead of products of labels, has been introduced by Chartrand et al in [1] and investigated by many authors (see e.g. [2][3][4]).…”

Section: Introductionmentioning

confidence: 99%

Product irregularity strength of graphs

Anholcer

2009

Discrete Mathematics

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a b s t r a c tConsider a simple graph G with no isolated edges and at most one isolated vertex. AThe goal is to obtain a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength. The analogous concept of irregularity strength, with sums in place of products, has been introduced by Chartrand et al. and investigated by many authors.

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Section: Introductionmentioning

confidence: 99%

Product irregularity strength of graphs

Anholcer

2009

Discrete Mathematics

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“…To overcome this issue, Chartrand et al proposed the following approach [5]: transform the graph G into some totally irregular multigraph G by replacing each edge e of G by a set of n e parallel edges, with n e ≥ 1. Since two vertices are adjacent in G if and only if they are adjacent in G, the structures of G and G are similar.…”

Section: Introductionmentioning

confidence: 99%

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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“…Colorings and labellings related to sums of the colors have been studied widely, see (Chartrand et al 1988;Kalkowski et al 2010;Przybyło 2009;Przybyło and Woźniak 2011;Wong and Zhu 2011). Among them there is the 1-2 Conjecture due to Przybyło and Woźniak (2010).…”

Section: Conjecture 12mentioning

confidence: 98%