Asymptotic enumeration by degree sequence of graphs of high degree
European Journal of Combinatorics
On the impact of sense of direction on message complexity
Information Processing Letters
Complexity of Deciding Sense of Direction
SIAM Journal on Computing
Sense of Direction in Distributed Computing
DISC '98 Proceedings of the 12th International Symposium on Distributed Computing
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A graph G with n vertices and maximum degree 驴G cannot be given weak sense of direction using less than 驴G colours. It is known that n colours are always sufficient, and it was conjectured that just 驴G + 1 are really needed, that is, one more colour is sufficient. Nonetheless, it has just been shown [2] that for sufficiently large n there are graphs requiring 驴(n/ log n) more colours than 驴G. In this paper, using recent results in asymptotic graph enumeration, we show not only that (somehow surprisingly) the same bound holds for regular graphs, but also that it can be improved to 驴(n log log n/ log n). We also show that 驴(dG驴log log dG) colours are necessary, where dG is the degree of G.