A k-star colouring of a graph G is a function
f:V(G)β{0,1,β¦,kβ1} such that f(u)ξ =f(v) for every edge uv of
G, and every bicoloured connected subgraph of G is a star. The star
chromatic number of G, Οsβ(G), is the least integer k such that G is
k-star colourable. We prove that Οsβ(G)β₯β(d+4)/2β for
every d-regular graph G with dβ₯3. We reveal the structure and
properties of even-degree regular graphs G that attain this lower bound. The
structure of such graphs G is linked with a certain type of Eulerian
orientations of G. Moreover, this structure can be expressed in the LC-VSP
framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by
an FPT algorithm with the parameter either treewidth, cliquewidth, or
rankwidth. We prove that for pβ₯2, a 2p-regular graph G is
(p+2)-star colourable only if n:=β£V(G)β£ is divisible by (p+1)(p+2). For
each pβ₯2 and n divisible by (p+1)(p+2), we construct a 2p-regular
Hamiltonian graph on n vertices which is (p+2)-star colourable.
The problem k-STAR COLOURABILITY takes a graph G as input and asks
whether G is k-star colourable. We prove that 3-STAR COLOURABILITY is
NP-complete for planar bipartite graphs of maximum degree three and arbitrarily
large girth. Besides, it is coNP-hard to test whether a bipartite graph of
maximum degree eight has a unique 3-star colouring up to colour swaps. For
kβ₯3, k-STAR COLOURABILITY of bipartite graphs of maximum degree k is
NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH
fails