Star colouring and locally constrained graph homomorphisms

We relate star colouring of even-degree regular graphs to the notions of locally constrained graph homomorphisms to the oriented line graph $\vec{L}(K_q)$ of the complete graph $K_q$ and to its underlying undirected graph $L^*(K_q)$. Our results have consequences for locally constrained graph homomorphisms and oriented line graphs in addition to star colouring. We show that $L^*(H)$ is a 2-lift of the line graph $L(H)$ for every graph $H$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal (J. Graph Theory, 2013) proved that for every 3-regular graph $G$, the line graph of $G$ is 4-star colourable if and only if $G$ admits a locally bijective homomorphism to the cube $Q_3$. We generalise this result as follows: for $p\geq 2$, a $K_{1,p+1}$-free $2p$-regular graph $G$ admits a $(p+2)$-star colouring if and only if $G$ admits a locally bijective homomorphism to $L^*(K_{p+2})$. As a result, if a $K_{p+1}$-free $2p$-regular graph $G$ with $p\geq 2$ is $(p+2)$-star colourable, then $-2$ and $p-2$ are eigenvalues of $G$. We also prove the following: (i) for $p\geq 2$, a $2p$-regular graph $G$ admits a $(p+2)$-star colouring if and only if $G$ has an orientation that admits an out-neighbourhood bijective homomorphism to $\vec{L}(K_{p+2})$; (ii) the line graph of a 3-regular graph $G$ is 4-star colourable if and only if $G$ 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.