Star colouring and locally constrained graph homomorphisms

Dvořák, Mohar and Šámal (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.