Upper bound for zero-sum Ramsey numbers of (k−1)-good graphs

Determine whether, for every integer k≥3 and every (k−1)-good graph G with k dividing e(G), the zero-sum Ramsey number satisfies R(G,Z_k)≤n+k−1.

Background

The paper defines a (k−1)-good graph as a graph containing at least k−1 degree-one vertices that are pairwise at distance at least three. Its modulo-3 arguments establish the analogous n+2 upper bound for 2-good graphs. The concluding problem asks whether the corresponding bound n+k−1 holds for all k≥3 and all (k−1)-good graphs with a number of edges divisible by k.