1 research outputs found
Star colouring and locally constrained graph homomorphisms
Dvo\v{r}\'ak, Mohar and \v{S}\'amal (J. Graph Theory, 2013) proved that for
every 3-regular graph , the line graph of is 4-star colourable if and
only if admits a locally bijective homomorphism to the cube . We
generalise this result as follows: for , a -free
-regular graph admits a -star colouring if and only if
admits a locally bijective homomorphism to a fixed -regular graph named
. We also prove the following: (i) for , a -regular graph
admits a -star colouring if and only if has an orientation
that admits an out-neighbourhood bijective homomorphism to a fixed
orientation of ; (ii) for every 3-regular graph , the
line graph of is 4-star colourable if and only if is bipartite and
distance-two 4-colourable; and (iii) it is NP-complete to check whether a
planar 4-regular 3-connected graph is 4-star colourable