On the injective chromatic number of graphs

Hahn, Geňa; Kratochvíl, Jan; Širáň‬, Jozef; Sotteau, Dominique

doi:10.1016/s0012-365x(01)00466-6

Cited by 123 publications

(69 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The injective coloring of graphs was first introduced by Hahn, Kratochvíl,Širáň and Sotteau [7] in 2002. The first results on the injective chromatic number of planar graphs were presented by Doyon, Hahn and Raspaud [6] in 2005.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples to a conjecture on injective colorings

Lužar¹,

Škrekovski

2015

Ars Math. Contemp.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Counterexamples to a conjecture on injective colorings

Lužar¹,

Škrekovski

2015

Ars Math. Contemp.

View full text Add to dashboard Cite

show abstract

“…In [4], interesting results were given for injective colorings of Cartesian graph products, especially hypercubes , where the results were stated as follows. …”

Section: Results For the Graph Productmentioning

confidence: 99%

“…The injective coloring of a graph was introduced by Hahn et al in [4], in which they proved the inequality ࣘ χ i (G) ࣘ 2 − + 1, where is the maximum degree of G. They characterized the graphs for which the bound is attained in the inequality. They also reported some interesting results on injective colorings of Cartesian graph products, especially of hypercubes.…”

Section: Introductionmentioning

confidence: 99%

Injective coloring of some graph operations

Song

Yue

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…Locally injective homomorphisms of undirected graphs are studied in [6][7][8][9][10]. In some cases they are referred to as partial covers, in contrast to full covers, which may be viewed as locally bijective homomorphisms (the mapping is bijective on neighbourhoods).…”

mentioning

confidence: 99%

“…This may be viewed as the least number of colours needed to colour the vertices of a graph such that the neighbours of a vertex all receive different colours, with the added condition that adjacent vertices may receive the same colour. Hahn et al [10] were particularly interested in determining the reflexive injective chromatic number of the hypercube because of its connections to coding theory. Since we are mostly interested in complexity results, this will be the only type of result from [10] that we will list here.…”

mentioning

confidence: 99%