Rainbow variations on a theme by mantel: extremal problems for Gallai colouring templates
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| Title: | Rainbow variations on a theme by mantel: extremal problems for Gallai colouring templates |
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| Authors: | Falgas-Ravry, Victor, Markström, Klas, Räty, Eero |
| Source: | Combinatorica. 44:997-1010 |
| Subject Terms: | 05C35, 05D99, Extremal graph theory, Gallai colourings, Mantel’s theorem, Rainbow triangles |
| Description: | Let G:=(G1,G2,G3) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in G is a triple of edges (e1,e2,e3) with ei ∈ Gi for each i and {e1,e2,e3} forming a triangle in V. The triples G not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities (α1,α2,α3) such that if |E(Gi)| >αin2 for each i and n is sufficiently large, then G must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu. |
| File Description: | electronic |
| Access URL: | https://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-224095 https://doi.org/10.1007/s00493-024-00102-6 |
| Database: | SwePub |
Nájsť tento článok vo Web of Science | Abstract: | Let G:=(G1,G2,G3) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in G is a triple of edges (e1,e2,e3) with ei ∈ Gi for each i and {e1,e2,e3} forming a triangle in V. The triples G not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities (α1,α2,α3) such that if |E(Gi)| >αin2 for each i and n is sufficiently large, then G must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu. |
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| DOI: | 10.1007/s00493-024-00102-6 |