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 a fixed $2p$-regular graph named $G_{2p}$. 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 $\vec{G}$ that admits an out-neighbourhood bijective homomorphism to a fixed orientation $\vec{G_{2p}}$ of $G2p$; (ii) for every 3-regular graph $G$, the line graph of $G$ 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.

翻译：Dvořák、Mohar和Šámal（《图论杂志》，2013年）证明：对于每个3-正则图$G$，其线图是4-星可着色的当且仅当$G$存在一个局部双射同态到立方体$Q_3$。我们将此结果推广如下：对于$p\geq 2$，一个无$K_{1,p+1}$的$2p$-正则图$G$存在一个$(p+2)$-星着色当且仅当$G$存在一个局部双射同态到一个固定的$2p$-正则图$G_{2p}$。我们还证明了以下结论：(i) 对于$p\geq 2$，一个$2p$-正则图$G$存在一个$(p+2)$-星着色当且仅当$G$存在一个定向$\vec{G}$，使得该定向存在一个出邻域双射同态到$G_{2p}$的固定定向$\vec{G_{2p}}$；(ii) 对于每个3-正则图$G$，其线图是4-星可着色的当且仅当$G$是二分图且距离为二时是4-可着色的；(iii) 判断一个平面4-正则3-连通图是否为4-星可着色是NP完全的。