Exact Values for the $ϵ$-Ascent Chromatic Index of Complete Graphs

Abstract

Following a problem introduced by Schurch [M. Schurch, $OntheDepressionofGraphs$, Doctoral Dissertation, University of Victoria, 2013], we find exact values of the minimum number of colours required to properly edge colour $K_n$, $n\ge 6$, 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 $\epsilon$-to appear-Ascent Chromatic Index of Complete Graphs, $Involve$, to appear].