Star colouring and locally constrained graph homomorphisms

Abstract

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.