Clustered Graph Coloring and Layered Treewidth

A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies clustered coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption than previous works, where typically the number of colors depends on an excluded minor. This paper focuses on graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, amongst other examples. Our main theorem says that for fixed integers $s,t,k$, every graph with layered treewidth at most $k$ and with no $K_{s,t}$ subgraph is $(s+2)$-colorable with bounded clustering. In the $s=1$ case, which corresponds to graphs of bounded maximum degree, we obtain polynomial bounds on the clustering. This greatly improves a corresponding result of Esperet and Joret for graphs of bounded genus. The $s=3$ case implies that every graph with a drawing on a fixed surface with a bounded number of crossings per edge is 5-colorable with bounded clustering. Our main theorem is also a critical component in two companion papers that study clustered coloring of graphs with no $K_{s,t}$-subgraph and excluding a fixed minor, odd minor or topological minor