A note on \(n\)-edge chromatic number (Q1387427)

All graphs \(G\) appearing in this note are simple. The maximum degree of \(G\) is denoted by \(\Delta(G)\). Let \(n\geq 2\) be an integer. The \(n\)-edge chromatic number \(\chi_n'(G)\) of a simple graph \(G\) is the minimum cardinality of a set of colors with which one can assign colors to the edges of \(G\) such that the edges on a path of length less than or equal to \(n\) receive different colors. The aim of this note is to explore bounds for \(\chi_n'(G)\) and \(\chi_n'(G)+ \chi_n'(\overline G)\).