For integers n, r, s, k ∈ N, n ≥ k and r ≥ s, let m (n, r, s, k) be the largest (in order) k-connected component with at most s colours one can find in anyr -colouring of the edges of the complete graph Kn on n vertices. Bollobás asked for the determination of m (n, r, s, k). Here, bounds are obtained in the cases s = 1, 2 and k = o (n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi's Regularity Lemma for many colours. We shall also study a similar question for bipartite graphs. © 2009 Elsevier B.V. All rights reserved.
Liu, H., & Person, Y. (2009). Highly connected coloured subgraphs via the regularity lemma. Discrete Mathematics, 309(21), 6277–6287. https://doi.org/10.1016/j.disc.2009.06.022