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تسجيل مستخدم جديدDefective and Clustered Choosability 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 $frac{2d+2}{d+2} k$ is $k$-choosable with defect $d$. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree $m$, no $(1-epsilon)m$ 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 $lfloor{frac{3}{4}m+1}rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree $m$ is $lfloor{frac{7}{10}m+1}rfloor$-choosable with clustering $9$, and is $lfloor{frac{2}{3}m+1}rfloor$-choosable with clustering $O(m)$. As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known 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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