Locally injective kk-colourings of planar graphs

A colouring of the vertices of a graph is called injective if every two distinct vertices connected by a path of length 2 receive different colours, and it is called locally injective if it is an injective proper colouring. We show that for k greater than or equal to 4k greater than or equal to 4, d...

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Bibliographic Details
Published inDiscrete Applied Mathematics Vol. 173; pp. 53 - 61
Main Authors Kratochvil, Jan, Siggers, Mark
Format Journal Article
LanguageEnglish
Published 01.08.2014
Subjects
Color
Colour
Colouring
Graphs
Mathematical analysis
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Summary:A colouring of the vertices of a graph is called injective if every two distinct vertices connected by a path of length 2 receive different colours, and it is called locally injective if it is an injective proper colouring. We show that for k greater than or equal to 4k greater than or equal to 4, deciding the existence of a locally injective kk-colouring, and of an injective kk-colouring, are NPNP-complete problems even when restricted to planar graphs. It is known that every planar graph of maximum degree less than or equal to 35k-52 allows a locally injective kk-colouring. To compare the behaviour of planar and general graphs we show that for general graphs, deciding the existence of a locally injective kk-colouring becomes NPNP-complete for graphs of maximum degree 2k (when k greater than or equal to 7k greater than or equal to 7).
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ISSN:0166-218X
DOI:10.1016/j.dam.2014.03.020