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Maximum number of triangle-free edge colourings with five and six colours

Botler, F., Corsten, J., Dankovics, A., Frankl, N., Hàn, H., Jiménez, A. and Skokan, J. (2019) Maximum number of triangle-free edge colourings with five and six colours. Acta Mathematica Universitatis Comenianae, 88 (3). pp. 495-499. ISSN 0862-9544

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Abstract

Let k ≥ 3 and r ≥ 2 be natural numbers. For a graph G, let F(G, k, r) denote the number of colourings of the edges of G with colours 1,…, r such that, for every colour c ∈ {1,…, r}, the edges of colour c contain no complete graph on k vertices Kk. Let F(n, k, r) denote the maximum of F(G, k, r) over all graphs G on n vertices. The problem of determining F(n, k, r) was first proposed by Erdős and Rothschild in 1974, and has so far been solved only for r = 2; 3, and a small number of other cases. In this paper we consider the question for the cases k = 3 and r = 5 or r = 6. We almost exactly determine the value F(n, 3, 6) and approximately determine the value F(n, 3, 5) for large values of n. We also characterise all extremal graphs for r = 6 and prove a stability result for r = 5.

Item Type: Article
Additional Information: © 2019 Acta Mathematica Universitatis Comenianae
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 08 Nov 2019 15:45
Last Modified: 20 Nov 2019 12:45
URI: http://eprints.lse.ac.uk/id/eprint/102464

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