an:07186363
Zbl 1436.05038
Hendrey, Kevin; Wood, David R.
Defective and clustered choosability of sparse graphs
EN
Comb. Probab. Comput. 28, No. 5, 791-810 (2019).
0963-5483 1469-2163
2019
j
05C15 05C42 05C07
choosability of graphs
Summary: 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 \textit{F. Havet} and \textit{J.-S. Sereni} [J. Graph Theory 52, No. 3, 181--199 (2006; Zbl 1104.05026)]. For clustered choosability of graphs with maximum average degree \(m\), no \((1-\varepsilon )m\) 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 \textit{M. Kopreski} and \textit{G. Yu} [Discrete Math. 340, No. 10, 2528--2530 (2017; Zbl 1367.05075)] 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.
1104.05026; 1367.05075