K5 is the only double-critical 5-chromatic graph

Abstract A connected graph G is called double-critical if the chromatic number of G decreases by two if any two adjacent vertices of G are removed. L. Lovasz conjectured that Kk is the only double-critical graph with chromatic number k. This is almost trivial for k⩽4 and the aim of this note is to prove this conjecture for k = 5.