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 ˜Θ n 1 r+1 (that is, n 1 r+1 up to a factor poly-logarithmic in 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.

• Type: Journal
• Year: 2016
• Where: DM
• Authors: Noga Alon, Omri Ben-Eliezer
• Keywords: DM 2016