Some structural properties of planar graphs and their applications to 3-choosability

Min Chen (1) , Mickael Montassier (1) , André Raspaud (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph $G$ is 3-choosable if it is satisfied one of the following conditions: (1) each vertex $x$ is neither incident to cycles of lengths $4,9,i_x$ with $i_x \in \{5,7,8\}$, nor incident to 6-cycles adjacent to a 3-cycle. (2) each vertex $x$ is not incident to cycles of lengths $4,7,9,i_x$ with $i_x\in \{5,6,8\}$.

Domaines: Mathématique discrète [cs.DM]

Min Chen, Mickael Montassier, André Raspaud. Some structural properties of planar graphs and their applications to 3-choosability. 2009. ⟨hal-00375596⟩