Asymptotics of the list-chromatic index for multigraphs

Randomly colouring graphs (a combinatorial view)

“…Hard-core distributions have been successfully used by Kahn [100,101] to prove the following strong result. Theorem 7.6 (Kahn, 2000). For multigraphs G, χ f (L (G)) ≃ ch ′ (G) as χ f (L (G)) → ∞.…”

“…Entropy was introduced by Shannon, and it plays a fundamental role in information theory. It has also proved to be a useful tool for some combinatorial problems, including graph colouring problems [100,101]. In particular, the ideas of the theory of information can be applied to study counting questions and graph covering issues [156].…”

Coloring the square of a planar graph

“…[8,9,11] ). In particular, using a sophisticated argument due to Kahn [8], we can prove the following lemma which specifies certain sets of removable vertices which we can use to perform reductions.…”

“…The proof of Lemma 2.2 is much more complicated. We follow the approach developed by Kahn [8] for his proof that the list chromatic index of a multigraph is asymptotically equal to its fractional chromatic number. We need to modify the proof so it can handle our situation in which we have a graph which is slightly more than a line graph and in which we have lists with fewer colours than he permitted.…”

Asymptotically optimal frugal colouring

“…Doing so requires an analysis of the probability of a set of vertices X all receiving the same colour, conditioned on the assignments of colours to vertices not in X. Conditioning on the assignments to vertices coloured before those in X is straightforward, but conditioning on assignments made after (or between) the vertices of X is the sort of thing that is often very problematic (see for example, Kahn's discussion in the epilogue of [14]). Fortunately, in this particular setting, we were able to handle the conditioning adequately.…”