Locally identifying colourings for graphs with given maximum degree

Florent Foucaud (1) , Iiro Honkala (2) , Tero Laihonen (3) , Aline Parreau (4) , Guillem Perarnau (5)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 Department of Mathematics
3 Turku Centre for Computer Science
4 IF - Institut Fourier
5 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]

A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that any graph G has a locally identifying colouring with $2\Delta^2-3\Delta+3$ colours, where $\Delta$ is the maximum degree of G, answering in a positive way a question asked by Esperet et al. We also provide similar results for locally identifying colourings which have the property that the colours in the neighbourhood of each vertex are all different and apply our method to the class of chordal graphs.

Mots clés: Graph colouring; Identification; Maximum degree; Chordal graphs

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Florent Foucaud, Iiro Honkala, Tero Laihonen, Aline Parreau, Guillem Perarnau. Locally identifying colourings for graphs with given maximum degree. Discrete Mathematics, 2012, 312 (10), pp.1832--1837. ⟨10.1016/j.disc.2012.01.034⟩. ⟨hal-00614513v2⟩