Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

Authors Zdenek Dvorák , Ken-ichi Kawarabayashi

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Zdenek Dvorák
  • Charles University, Malostranske namesti 25, 11800 Prague, Czech Republic
Ken-ichi Kawarabayashi
  • National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

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Zdenek Dvorák and Ken-ichi Kawarabayashi. Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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 K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k >= 1 and 1 <=beta <=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)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. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • non-approximability
  • chromatic number
  • minor-closed classes


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