Star Colouring of Regular Graphs Meets Weaving and Line Graphs

For q is an element of N, a q-star colouring of a graph G is a proper q-colouring f of G such that there is no path u, v, w, x in G with f(u) = f(w) and f(v) = f(x) (the violating path need not be induced). For p >= 2, Shalu and Antony (Discrete Math., 2022) proved that at least p + 2 colours are required to star colour a 2p-regular graph G, and characterised the class G of graphs G for which p+ 2 colours suffices in terms of graph orientations. In the second author's thesis (2023), we provided a characterisation of the class G in terms of locally constrained graph homomorphisms. In this paper, we characterise G in terms of weaving patterns of edge decompositions. We also show that the study of class G is tied to the theory of line graphs and line digraphs of complete graphs. We prove that if a K-1,K- p+1-free 2p-regular graph G with p >= 2 is (p + 2)-star colourable, then -2 and p-2 are eigenvalues of the adjacency matrix of G.