Local chromatic number and the Borsuk-Ulam Theorem

The local chromatic number of a graph was introduced in [13]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the stable Kneser (or Schrijver) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. Our method for the lower bounds on the local chromatic number is topological. We identify “topological reasons” for a graph to be at least t-chromatic. The above mentioned graphs satisfy these conditions if t is their chromatic number. We show that these conditions are enough to find t distinct colors “locally” (in a complete bipartite subgraph) in any proper coloring with an arbitrary number of colors. This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs, which is tight or almost tight in many cases. As a consequence of our results we find a generalization of (the LyusternikSchnirel’man version of) the Borsuk-Ulam Theorem. As another consequence, we also prove a conjecture of Johnson, Holroyd, and Stahl on the circular chromatic number of Kneser graphs in the case the chromatic number of the graph is even.