Exact Values for the \(\)-Ascent Chromatic Index of Complete Graphs

Following a problem introduced by Schurch [M. Schurch, \emph{On the Depression of Graphs}, Doctoral Dissertation, University of Victoria, 2013], we find exact values of the minimum number of colours required to properly edge colour \( K_n \), \( n \geq 6 \), using natural numbers, such that the length of a shortest maximal path of increasing edge labels is equal to three. This result improves the result of Breytenbach and Mynhardt [A. Breytenbach and C. M. Mynhardt, On the \(\varepsilon\)-to appear-Ascent Chromatic Index of Complete Graphs, \emph{Involve}, to appear].