Geodesic
A rainbow version of Mantel's Theorem
Advances in Combinatorics (2020)
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Mantel's Theorem asserts that a simple $n$ vertex graph with more than $\frac{1}{4}n^2$ edges has a triangle (three mutually adjacent vertices). Here we consider a rainbow variant of this problem. We prove that whenever $G_1, G_2, G_3$ are simple graphs on a common set of $n$ vertices and $|E(G_i)| > ( \frac{ 26 - 2 \sqrt{7} }{81})n^2 \approx 0.2557 n^2$ for $1 \le i \le 3$, then there exist distinct vertices $v_1,v_2,v_3$ so that (working with the indices modulo 3) we have $v_i v_{i+1} \in E(G_i)$ for $1 \le i \le 3$. We provide an example to show this bound is best possible. This also answers a question of Diwan and Mubayi. We include a new short proof of Mantel's Theorem we obtained as a byproduct.
Publié le : 
@article{ADVC_2020_a10,
     author = {Ron Aharoni and Matt DeVos and Sebasti\'an Gonz\'alez Hermosillo de la Maza and Amanda Montejano and Robert \v{S}\'amal},
     title = {A rainbow version of {Mantel's} {Theorem}},
     journal = {Advances in Combinatorics},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADVC_2020_a10/}
}
TY  - JOUR
AU  - Ron Aharoni
AU  - Matt DeVos
AU  - Sebastián González Hermosillo de la Maza
AU  - Amanda Montejano
AU  - Robert Šámal
TI  - A rainbow version of Mantel's Theorem
JO  - Advances in Combinatorics
PY  - 2020
UR  - http://geodesic.mathdoc.fr/item/ADVC_2020_a10/
LA  - en
ID  - ADVC_2020_a10
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%0 Journal Article
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%A Matt DeVos
%A Sebastián González Hermosillo de la Maza
%A Amanda Montejano
%A Robert Šámal
%T A rainbow version of Mantel's Theorem
%J Advances in Combinatorics
%D 2020
%U http://geodesic.mathdoc.fr/item/ADVC_2020_a10/
%G en
%F ADVC_2020_a10
Ron Aharoni; Matt DeVos; Sebastián González Hermosillo de la Maza; Amanda Montejano; Robert Šámal. A rainbow version of Mantel's Theorem. Advances in Combinatorics (2020). http://geodesic.mathdoc.fr/item/ADVC_2020_a10/