Asymptotics of the list-chromatic index for multigraphs
Random Structures & Algorithms
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A new bound on the cyclic chromatic number
Journal of Combinatorial Theory Series B
Choosability conjectures and multicircuits
Discrete Mathematics
Note: coloring the square of a K4-minor free graph
Discrete Mathematics
Coloring Powers of Planar Graphs
SIAM Journal on Discrete Mathematics
A bound on the chromatic number of the square of a planar graph
Journal of Combinatorial Theory Series B
Coloring the square of a planar graph
Journal of Graph Theory
A new upper bound on the cyclic chromatic number
Journal of Graph Theory
Facial colorings using Hall's Theorem
European Journal of Combinatorics
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We introduce the notion of (A, B)-colouring of a graph: For given vertex sets A, B, this is a colouring of the vertices in B so that both adjacent vertices and vertices with a common neighbour in A receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of plane graphs. We prove a general result which implies asymptotic versions of Wegner's and Borodin's Conjecture on these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs and then use Kahn's result that the list chromatic index is close from the fractional chromatic index. Our results are based on a strong structural lemma for planar graphs which also implies that the size of a clique in the square of a planar graph of maximum degree Δ is at most 3/2 Δ plus a constant.