Quadrangulations and 4-color-critical graphs

Erdős asked if the removal of few edges in a large 4-color-critical graph always leaves a 3-chromatic graph. Erdős and Hajnal asked if a graph is 3-colorable if all its induced subgraphs can be made bipartite by the omission of few edges (relative to the number of vertices). We answer both problems, which are stated in Bollobas’ monograph Extremal Graph Theory from 1978, in the negative.