On the dynamic coloring of graphs

Alishahi, Meysam

doi:10.1016/j.dam.2010.10.012

Cited by 49 publications

(33 citation statements)

References 7 publications

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“…For example, Dehghan, and Ahadi [8] showed that if G is regular, then χ 2 (G) ≤ χ(G) + 2 log α(G) + 3, where α is the independence number of the graph. Alishahi [4] showed that if G is d-regular,…”

Section: Introductionmentioning

confidence: 99%

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

Bowler

Erde

Lehner

et al. 2017

Discrete Applied Mathematics

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Abstract.A dynamic colouring of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The dynamic colouring number χ 2 (G) of a graph G is the least number of colours needed for a dynamic colouring of G.Montgomery conjectured that χ 2 (G) ≤ χ(G) + 2 for all regular graphs G, which would significantly improve the best current upper bound χ 2 (G) ≤ 2χ(G). In this note, however, we show that this last upper bound is sharp by constructing, for every integer n ≥ 2, a regular graph G with χ(G) = n but χ 2 (G) = 2n. In particular, this disproves Montgomery's conjecture.

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Section: Introductionmentioning

confidence: 99%

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

Bowler

Erde

Lehner

et al. 2017

Discrete Applied Mathematics

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“…Oum [16] in 2016. For other results on the dynamic coloring of graphs, we refer the reads to [1,4,8,9,10,13,18,19,20,21,24,26].…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, Esperet [12] showed that there is a planar bipartite graph G with ch(G) = χ d (G) = 3 and ch d (G) = 4, and moreover, there exists for every k ≥ 5 a bipartite graph G k with ch(G k ) = χ d (G k ) = 3 and ch d (G k ) ≥ k. Hence the gap between χ d (G) (or ch(G)) and ch d (G) can be any large. For further interesting readings on the dynamic list coloring of graphs, we refer the readers to [4,14,15]. Given a graph G, we insert on each edge a new vertex of degree two, and call the resulting graph, denoted by G , a 2-subdivision.…”

Section: Introductionmentioning

confidence: 99%

List edge and list total coloring of 1-planar graphs

2012

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A graph is k-planar if it can be drawn in the plane so that each edge is crossed at most k times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called "beyond planar graphs". A dynamic -list coloring of a graph is a proper coloring so that each vertex receives a color from a list of distinct candidate colors assigned to it, and meanwhile, there are at least two colors appearing in the neighborhood of every vertex of degree at least two. In this paper, we prove that each 1-planar graph has a dynamic 11-list coloring. Moreover, we show a relationship between the dynamic coloring of 1-planar graphs and the proper coloring of 2-planar graphs, which states that the dynamic (list) chromatic number of the class of 1-planar graphs is at least the (list) chromatic number of the class of 2-planar graphs.

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“…The 1-dynamic chromatic number of a graph G is 1(G) = (G), well-known as the ordinary chromatic number of G. The 2-dynamic chromatic number is simply said to be a dynamic chromatic number, denoted by2(G)= d(G),see Montgomery [4]. The r-dynamic chromatic number has been studied by several authors, for instance in [1], [5], [6], [7], [8], [10], [11]. The following observations are useful for our study, proposed by Jahanbekam [11].…”

Section: Introductionmentioning

confidence: 99%

On r-Dynamic Chromatic Number of the Corronation of Path and Several Graphs

Kristiana¹,

Dafik²,

Utoyo³

et al. 2017

IJAERS

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Abstract-This study is a natural extension of k -proper coloring of any simple and connected graph G. By a n rdynamic coloring of a graph G, we mean a proper kcoloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. The r-dynamicchromatic number, written as r(G), is the minimum k such that graph G has an r-dynamic k -coloring. In this paper we will study the r-dynamic chromatic number of the coronation of path and several graph. We denote the corona product of G and H by ⨀ . We will obtain the rdynamic chromatic number of ( ⨀ ) , ( ⨀ ) and ( ⨀ ) for m, n 3.Keyword-r-dynamic chromatic number, path, corona product.

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On the dynamic coloring of graphs

Cited by 49 publications

References 7 publications

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

List edge and list total coloring of 1-planar graphs

On r-Dynamic Chromatic Number of the Corronation of Path and Several Graphs

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