  We show that the discrete versions of the systolic inequality that estimate
the number of vertices of a simplicial complex from below have substantial
applications to graphs, the one-dimensional simplicial complexes. Almost
directly they provide good estimates for the number of vertices of a graph in
terms of its chromatic number and the length of the smallest odd cycle.
Combined with the graph-theoretic techniques of Berlov and Bogdanov, the
systolic approach produces even better estimates.
