Edge colouring of \(K_{2n}\) with spanning star-forests receiving distinct colours (Q1300004)

The authors prove that for any spanning star-forest \(S= S_{n_1} \cup\cdots \cup S_{n_k}\neq S_{2n-3} \cup S_1\) or \(2S_2\) (if \(n=3)\) of \(K_{2n}\), where \(n_i\geq 1\) for all \(i=1,2, \dots, k\), there exists an edge-colouring of \(K_{2n}\) using \(2n-1\) colours such that all the edges of \(S\) receive distinct colours. The proof given here is constructive and can be used for finding a required edge colouring. This result is useful in the study of total colourings of graphs (colourings of the adjacent and incident vertices and edges).