Adding Edges to Increase the Chromatic Number of a Graph

If n ≤ k + 1 and G is a connected n-vertex graph, then one can add edges to G so that the resulting graph contains the complete graph K k+1. This yields that for any connected graph G with at least k + 1 vertices, one can add edges to G so that the resulting graph has chromatic number > k. A long time ago, Bollobás suggested that for every k ≤ 3 there exists a k-chromatic graph Gk such that after adding to it any-1 edges, the chromatic number of the resulting graph is still k. In this note we prove this conjecture.