Defective Colouring of Graphs Excluding A Subgraph or Minor

Abstract

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be 3-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no Kt+1-minor can be t-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.

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Correspondence to Patrice Ossona De Mendez.

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Ossona de Mendez is supported by grant ERCCZ LL-1201 and by the European Associated Laboratory "Structures in Combinatorics" (LEA STRUCO), and partially supported by ANR project Stint under reference ANR-13-BS02-0007.

Oum is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2017R1A2B4005020).

Wood is supported by the Australian Research Council.

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Ossona De Mendez, P., Oum, S. & Wood, D.R. Defective Colouring of Graphs Excluding A Subgraph or Minor. Combinatorica 39, 377–410 (2019). https://doi.org/10.1007/s00493-018-3733-1

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Mathematics Subject Classification (2010)

  • 05C15
  • 05C83
  • 05C10