A rainbow version of Mantel's Theorem
Advances in Combinatronics (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 Combinatronics},
publisher = {mathdoc},
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 Combinatronics PY - 2020 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ADVC_2020_a10/ LA - en ID - ADVC_2020_a10 ER -
Ron Aharoni; Matt DeVos; Sebastián González Hermosillo de la Maza; Amanda Montejano; Robert Šámal. A rainbow version of Mantel's Theorem. Advances in Combinatronics (2020). http://geodesic.mathdoc.fr/item/ADVC_2020_a10/