  This paper is concerned with sampling from the uniform distribution on
H-colourings of the n-vertex path using systematic scan Markov chains. An
H-colouring of the n-vertex path is a homomorphism from the n-vertex path to
some fixed graph H. We show that systematic scan for H-colourings of the
n-vertex path mixes in O(log n) scans for any fixed H. This is a significant
improvement over the previous bound on the mixing time which was O(n^5) scans.
Furthermore we show that for a slightly more restricted family of H (where any
two vertices are connected by a 2-edge path) systematic scan also mixes in
O(log n) scans for any scan order. Finally, for completeness, we show that a
random update Markov chain mixes in O(n log n) updates for any fixed H,
improving the previous bound on the mixing time from O(n^5) updates.
