Local and global colorability of graphs

It is shown that for any fixed c >= 3 and r, the maximum possible chromatic number of a graph on n vertices in which every subgraph of radius at most r is c-colorable is (Theta) over tilde (n 1/r+1): it is O ((n/ log n) 1/r+1) and Omega(n 1/r+1 log n). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random n-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius r in it are 2-degenerate. (c) 2015 Elsevier B.V. All rights reserved.