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 $\binom{k}{2}$ 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.
2010 ◽
Vol 02
(02)
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pp. 207-211
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1972 ◽
Vol 24
(5)
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pp. 805-807
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2010 ◽
Vol 21
(03)
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pp. 311-319
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2012 ◽
Vol 12
(02)
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pp. 1250151
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