  We consider two different notions of graph colouring, namely, the
$t$-periodic colouring for vertices that has been introduced in 1974 by Bondy
and Simonovits, and the periodic colouring for oriented edges that has been
recently introduced in the context of spectral theory of non-backtracking
operators. For each of these two colourings, we introduce the corresponding
colouring number which is given by maximising the possible number of colours.
We first investigate these two new colouring numbers individually, and we then
show that there is a deep relationship between them.
