Lower bounds for sense of direction in regular graphs

Abstract.

A graph G with n vertices and maximum degree $\Delta_G$ cannot be given weak sense of direction using less than $\Delta_G$ colours. It is known that n colours are always sufficient, and it was conjectured that just $\Delta_G + 1$ are really needed, that is, one more colour is sufficient. Nonetheless, it has been shown [3] that for sufficiently large n there are graphs requiring $\Omega(n\log\log n/\log n)$ more colours than $\Delta_G$. In this paper, using recent results in asymptotic graph enumeration, we show that (surprisingly) the same bound holds for regular graphs. We also show that $\Omega\left(d_G\sqrt{\log\log d_G}\right)$ colours are necessary, where d _G is the degree of G.