Let G be a toroidal graph without cycles of a fixed length k, and l (G) the list chromatic number of G. We establish tight upper bounds Contract grant sponsor: RGC; Contract grant number: CUHK4165/01E (to L. C.); Contract grant sponsor: NSFC; Contract grant number: 10771197 (to W. W.); Contract grant sponsor: ZJNSFC; Contract grant number: Z6090150 (to W. W.); Contract grant sponsor: NSC97-2115...

We model a problem proposed by Alcatel, a satellite building company, using improper colourings of graphs. The relation between improper colourings and maximum average degree is underlined, which contributes to generalise and improve previous known results about improper colourings of planar graphs.

It is proved that the choice number of every graph G embedded on a surface of Euler genus and is at most the Heawood number H b p c and that the equality holds if and only if G contains the complete graph KH as a subgraph