On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1609.gif" display="inline" overflow="scroll"><mml:mi>r</mml:mi></mml:math>-hued coloring of planar graphs with girth at least 6

Song, Huimin; Lai, Hong‐Jian; Jian, Wu

doi:10.1016/j.dam.2015.05.015

Cited by 27 publications

(10 citation statements)

References 7 publications

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“…, where r ≥ 1. The notion of r-dynamic coloring was introduced by Montgomery [30], newly studied [19,18], and also investigated under the notion of r-hued coloring [11,27,31,32,33,41]. As starting cases of r-dynamic coloring, the 2-dynamic coloring (known as dynamic coloring in literature) [4,1,2,3,8,10,23,26,29,28] and the 3-dynamic coloring [5,21,25] have been considered.…”

Section: List 3-dynamic Coloringmentioning

confidence: 99%

The structure and the list 3-dynamic coloring of outer-1-planar graphs

Li,

Zhang

2019

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An outer-1-planar graph is a graph admitting a drawing in the plane for which all vertices belong to the outer face of the drawing and there is at most one crossing on each edge. This paper describes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of the other sixteen configurations. There are two interesting applications of this structural theorem.First of all, we conclude that every (resp. maximal) outer-1-planar graph with minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

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Section: List 3-dynamic Coloringmentioning

confidence: 99%

The structure and the list 3-dynamic coloring of outer-1-planar graphs

Li,

Zhang

2019

Preprint

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“…Lai et al [14] showed the conjecture 1.1 is true for planar graphs with girth at least 6. Several results about r-dynamic coloring were published in [1,2,5,6,7,11].…”

Section: Introductionmentioning

confidence: 99%

On list 3-dynamic coloring of near-triangulations

Kim

et al. 2019

Preprint

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An r-dynamic k-coloring of a graph G is a proper k-coloring such that for any vertex v, there are at least min{r, deg G (v)} distinct colors in N G (v). The r-dynamic chromatic number χ d r (G) of a graph G is the least k such that there exists an r-dynamic k-coloring of G. The list r-dynamic chromatic number of a graph G is denoted by ch d r (G). Loeb et al. [11] showed that ch d 3 (G) ≤ 10 for every planar graph G, and there is a planar graph G with χ d 3 (G) = 7. In this paper, we study a special class of planar graphs which have better upper bounds of ch d 3 (G). We prove that ch d 3 (G) ≤ 6 if G is a planar graph which is near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.

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“…And Cheng et al [3] studied the maximum average condition to have χ d 3 (G) ≤ 4, 5, or 6. On the other hand, Song et al [14] showed that if G is planar with girth at least 6, then χ d r (G) ≤ r + 5 for any r ≥ 3. In this paper, we study list 3-dynamic coloring in terms of maximum average degree.…”

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confidence: 99%