Some bounds on the injective chromatic number of graphs

Doyon, Alain; Hahn, Geňa; Raspaud, André

doi:10.1016/j.disc.2009.04.020

Cited by 35 publications

(13 citation statements)

References 2 publications

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“…The first results on the injective chromatic number of planar graphs were presented by Doyon, Hahn and Raspaud [6] in 2005. As a corollary of the main theorem they obtained that if G is a planar graph of maximum degree ∆ and girth g(G) ≥ 7, then the injective chromatic number is at most ∆ + 3.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples to a conjecture on injective colorings

Lužar¹,

Škrekovski

2015

Ars Math. Contemp.

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An injective coloring of a graph is a vertex coloring where two vertices receive distinct colors if they have a common neighbor. Chen, Hahn, Raspaud, and Wang [3] conjectured that every planar graph with maximum degree ∆ ≥ 3 admits an injective coloring with at most 3∆/2 colors. We present an infinite family of planar graphs showing that the conjecture is false for graphs with small or even maximum degree. We conclude this note with an alternative conjecture, which sheds some light on the well-known Wegner's conjecture for the mentioned degrees.

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Section: Introductionmentioning

confidence: 99%

Counterexamples to a conjecture on injective colorings

Lužar¹,

Škrekovski

2015

Ars Math. Contemp.

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“…Hahn, Raspaud and Wang [9] proved that the injective chromatic number of every K 4 -minor free graph of maximum degree ∆ is at most 3 2 ∆ . They also posed the following conjecture:…”

Section: Corollarymentioning

confidence: 99%

Injective colorings of planar graphs with few colors

Luar

Škrekovski

Tancer

2009

Discrete Mathematics

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a b s t r a c tAn injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥ 19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥ 10 are injectively (∆ + 1)-colorable, that ∆ + 4 colors are sufficient for planar graphs of girth ≥ 5 if ∆ is large enough, and that subcubic planar graphs of girth ≥ 7 are injectively 5-colorable.

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“…Theorem 2. Let G be a planar graph with maximum degree ∆(G) ≥ D and girth g(G) ≥ g. Then (a) (Borodin, Ivanova, Neustroeva [5]) if (D, g) ∈ { (3,24), (4,15), (5,13), (6,12), (7,11), (9,10), (15,8), (30, 7)}, then χ i (G) ≤ χ(G 2 ) = ∆ + 1.…”

mentioning

confidence: 99%