On the Asymptotics of Colouring Plane Multigraphs

For loopless plane multigraphs \(G\), the edge-face chromatic number and the entire chromatic number are asymptotically their fractional counterparts (LP relaxations) as these latter invariants tend to infinity. Proofs of these results are based on analogous theorems for the chromatic index and the total chromatic number, due, respectively, to Kahn [3] and to the first author [6]. Our two results fill in the missing pieces of a complete answer to the natural question: which of the seven invariants associated with colouring the nonempty subsets of \(\{V, E, F\}\) exhibit “asymptotically good” behaviour?