Vertex colorings of graphs without short odd cycles

Abstract

Motivated by the work of Nesetřil and Rodl on “Partitions of vertices” we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r-coloring of the vertices of H yields a monochromatic and induced copy of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 255-264, 2011 © 2011 Wiley Periodicals, Inc.