Georgia Institute of Technology College of Sciences

A graph is a minor of another graph if the former can be obtained from a subgraph of the latter by contracting edges. We prove that for every graph H, if H is not a minor of a graph G, then V(G) can be 3-colored such that the subgraph induced by each color class has no component with size greater than a function of H and the maximum degree of G. This answers a question raised by Esperet and Joret, generalizes their result for 3-coloring V(G) for graphs G embeddable in a fixed surface, and improves a result of Alon, Ding, Oporowski and Vertigan for 4-coloringing V(G) for H-minor free graphs G. As a corollary, we prove that for every positive integer t, if G is a graph with no K_{t+1} minor, then V(G) can be 3t-colored such that the subgraph induced by each color class has no component with size larger than a function of t. This improves a result of Wood for coloring V(G) by 3.5t+2 colors. This work is joint with Sang-il Oum.