Additive non-approximability of chromatic number in proper minor-closed classes - Dvořák, Zdeněk; Kawarabayashi, Ken-ichi (Journal of combinatorial theory. Series B, 2023)
Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally, for every k≥1 and 1≤β≤4/3, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4−3β)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. Furthermore, we give somewhat weaker non-approximability bound for K4k+1-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.