Bound on the number of colours in each extremal layer

Prove that in an optimal fundamental block for the k-colourable diameter problem, one may assume c(i) ≤ k − 1 for every layer i, where c(i) denotes the number of colour classes present in layer i.

Background

The paper studies optimal fundamental blocks arising from repeatable layered graphs for the maximum-diameter problem. For a k-colourable graph, c(i) records how many colours occur in the ith layer.

The authors state that the inequality c(i) ≤ k − 1 appears natural for extremal graphs but has not been established. They require this assumption to make the computational search tractable for δ = 16, so proving it would remove an important unverified restriction from their computational analysis.