Largest planar graphs of diameter two and fixed maximum degree
Discrete Mathematics
Asymptotics of the chromatic index for multigraphs
Journal of Combinatorial Theory Series B
Asymptotics of the list-chromatic index for multigraphs
Random Structures & Algorithms
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
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
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
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Graph Theory
Distance-two coloring of sparse graphs
European Journal of Combinatorics
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In this paper we introduce the notion of Σ-colouring of a graph G: For given subsets Σ(v) of neighbours of v, for every v驴V (G), this is a proper colouring of the vertices of G such that, in addition, vertices that appear together in some Σ(v) receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of 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 to the fractional chromatic index.Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree Δ embeddable in some fixed surface is at most $$ \frac{3} {2}\Delta $$ plus a constant.