Publication at Faculty of Mathematics and Physics |
2016
Abstract
If n >= k + 1 and G is a connected n-vertex graph, then one can add k(k-1)/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 k(k-1)/2 edges to G so that the resulting graph has chromatic number > k.
A long time ago, Bollobas suggested that for every k >= 3 there exists a k-chromatic graph G(k) such that after adding to it any k(k-1)/2 - 1 edges, the chromatic number of the resulting graph is still k. In this note we prove this conjecture.