An optimal square coloring of planar graphs

Bu, Yuehua; Zhu, Xubo

doi:10.1007/s10878-011-9409-z

Cited by 35 publications

(24 citation statements)

References 11 publications

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“…Since c 6 (u 5 ) = c 2 (u 5 ) ̸ = c 1 (x 1 ) = c 6 (x 1 ) and since |c 6 [x 1 ] ∪ c 6 [u 5 ]| ≤ r + 3 < k, it follows by Lemma 3.2 that c 6 can be extended to a (k, r)-coloring of G, contrary to (2). This proves Claim 5.…”

Section: Claim 2 For Anymentioning

confidence: 59%

“…Hence we must have c(v) ̸ = c(x). Since |c[x] ∪ c[v]| ≤ r + 4 < k, it follows by Lemma 3.2 that there exists a (k, r)-coloring c 4 of G, contrary to (2). This proves (i).…”

Section: Notations and Terminologymentioning

confidence: 61%

“…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: 82%

“…(i) If G is disconnected, then by (2), every component of G has a (k, r)-coloring, and so G has a (k, r)-coloring, contrary to (2). Hence G is connected.…”

Section: Notations and Terminologymentioning

confidence: 99%

“…Then we first extend c 0 to a partial (k, r)-coloring c 1 by letting c 1 (v) = a. Next apply Lemma 3.2 to extend c 1 to a partial (k, r)-coloring c 2 by coloring 5 [v]| ≤ 4 + 5 < 10, and so c 5 can be extended to a (k, r)-coloring c 6 of G by letting c 6 5 [v]| < 10, and so c 5 can always be extended to a (k, r)-coloring c 6 of G, contrary to (2). This proves Claim 1.…”

Section: Notations and Terminologymentioning

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: Claim 2 For Anymentioning

confidence: 59%

“…Hence we must have c(v) ̸ = c(x). Since |c[x] ∪ c[v]| ≤ r + 4 < k, it follows by Lemma 3.2 that there exists a (k, r)-coloring c 4 of G, contrary to (2). This proves (i).…”

Section: Notations and Terminologymentioning

confidence: 61%

Section: Conjecture 13 Let G Be a Planar Graph Thenmentioning

confidence: 82%

“…(i) If G is disconnected, then by (2), every component of G has a (k, r)-coloring, and so G has a (k, r)-coloring, contrary to (2). Hence G is connected.…”

Section: Notations and Terminologymentioning

confidence: 99%