Graph coloring with no large monochromatic components

Abstract

Abstract For a graph G and an integer t we let m c c t ( G ) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc 2 ( G ) = O ( n 2 / 3 ) for any n-vertex graph G ∈ F . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F , and every fixed t we show that mcc t ( G ) = O ( n 2 / ( t + 1 ) ) . On the other hand, we have examples of graphs G with no K t + 3 minor and with mcc t ( G ) = Ω ( n ( 2 / 2 t − 1 ) ) . It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved mcc 2 ( G ) ⩽ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc 2 ( G ) = Ω ( n ) , and more sharply, for every e > 0 there exists c e > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + e for all subgraphs, and with mcc 2 ( G ) ⩾ c e n . For 6-regular graphs it is known only that the maximum order of magnitude of mcc 2 is between n and n. We also offer a Ramsey-theoretic perspective of the quantity m c c t ( G ) .