Adding Edges to Increase the Chromatic Number of a Graph

Abstract

If n ⩾ k + 1 and G is a connected n-vertex graph, then one can add $\binom{k}{2}$ 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 $\binom{k}{2}$ 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 G_k such that after adding to it any $\binom{k}{2}$ − 1 edges, the chromatic number of the resulting graph is still k. In this note we prove this conjecture.