A vertex critical graph without critical edges

Abstract

A vertex k-critical graph is a k-chromatic graph with the property that the removal of any vertex leaves a ( k - 1)-colourable graph. A critical edge in a graph is an edge whose deletion lowers the chromatic number. Dirac conjectured in 1970 that for some k⩾4 there is a vertex k-critical graph with no critical edge. We produce here an example of such a graph for k = 5.