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Adding Edges to Increase the Chromatic Number of a Graph

Published online by Cambridge University Press:  31 March 2016

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Sobolev Institute of Mathematics, Novosibirsk, Russia (e-mail:
Computer Science Institute of Charles University, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, Praha 1, Czech Republic (e-mail


If nk + 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 Kk+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 Gk 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.

Copyright © Cambridge University Press 2016 

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