On dynamic coloring for planar graphs and graphs of higher genus

Chen, Ye; Fan, Shuxiang; Lai, Hong‐Jian; Song, Huimin; Sun, Lei

doi:10.1016/j.dam.2012.01.012

Cited by 38 publications

(23 citation statements)

References 7 publications

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“…Thus Theorem 1.4, together with Theorem 1.1 of [3] with 1 ≤ r ≤ 2, justifies Conjecture 1.3 for all planar graphs with girth at least 6. Bu and Zhu in [2] proved the special case when r = ∆ of Theorem 1.4, and so Theorem 1.4 is a generalization of this former result in [2].…”

Section: Conjecture 13 Let G Be a Planar Graph Thenmentioning

confidence: 52%

“…Now S(c 2 ) = V (G) − {s, t}. As c 2 (v 2 ) ̸ = c 2 (v 3 ), {c 2 (t)} = φ and as |c 2 [v 2 ] ∪ c 2 [v 3 ] ∪ c 2 [t]| ≤ 7, we conclude that c 2 can be extended to a partial (k, r)-coloring c 3 by defining c 3 …”

Section: H Song Et Al / Discrete Applied Mathematics ( ) -mentioning

confidence: 93%

“…Let c 0 be the restriction of c to 3 By (5), c 2 is a partial (k, r)-coloring with S(c 2 ) = V (G) − {t 1 , t 2 } such that c 2 (t ′ 1 ) ̸ = c 2 (v 4 ) and c 2 (t ′ 2 ) ̸ = c 2 (u 4 ). By Lemma 3.2, c 2 can be extended to a (k, r)-coloring of G, contrary to (2).…”

Section: H Song Et Al / Discrete Applied Mathematics ( ) -mentioning

confidence: 99%

“…In [6], an analogue of Brooks Theorem for χ 2 was proved. It was shown in [3] that χ 2 (G) ≤ 5 holds for any planar graph G. A Moore graph is a regular graph with diameter d and girth 2d + 1. Ding et al [4] proved that χ r (G) ≤ ∆ 2 + 1, where equality holds if and only if G is a Moore graph, which was improved to r∆ + 1 in [8].…”

Section: Introductionmentioning

confidence: 99%

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On r-hued coloring of planar graphs with girth at least 6

Song

Lai

Jian

2016

Discrete Applied Mathematics

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a b s t r a c tFor integers k, r > 0, a (k, r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v), r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χ r (G), is the least integer k such that a (k, r)-coloring of G exists. In this paper, we proved that if G is a planar graph with girth at least 6, then χ r (G) ≤ r + 5. This extends a former result in Bu and Zhu (2012). It also implies that a conjecture on r-hued coloring of planar graphs is true for planar graphs with girth at least 6.

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Section: Conjecture 13 Let G Be a Planar Graph Thenmentioning

confidence: 52%

Section: H Song Et Al / Discrete Applied Mathematics ( ) -mentioning

confidence: 93%

Section: H Song Et Al / Discrete Applied Mathematics ( ) -mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On r-hued coloring of planar graphs with girth at least 6

Song

Lai

Jian

2016

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%

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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Channel Assignment with r-Dynamic Coloring

Zhu

2018

Lecture Notes in Computer Science

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On dynamic coloring for planar graphs and graphs of higher genus

Cited by 38 publications

References 7 publications

On r-hued coloring of planar graphs with girth at least 6

On r-hued coloring of planar graphs with girth at least 6

List edge and list total coloring of 1-planar graphs

Channel Assignment with r-Dynamic Coloring

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