Defective and Clustered Colouring of Sparse Graphs
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An (improper) graph colouring has "defect" d if each monochromatic subgraph has maximum degree at most d, and has "clustering" c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than 2d+2/d+2 k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered colouring of graphs with maximum average degree m, no (1-ϵ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is 3/4m+1-choosable with clustering 2. We then prove a series of results for clustered colouring that explore the trade-off between the number of colours and the clustering. For example, we prove that every graph with maximum average degree m is 2/3m+1-choosable with clustering O(m). As an example, this implies that every biplanar graph is 8-choosable with bounded clustering. This is the first non-trivial result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.
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