Large Monochromatic Components of Small Diameter
Abstract
Gyárfás conjectured in 2011 that every $r$edgecolored $K_n$ contains a monochromatic component of bounded ("perhaps three") diameter on at least $n/(r1)$ vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of $r=3$: We show that in every $3$edgecoloring of $K_n$ either there is a monochromatic component of diameter at most three on at least $n/2$ vertices or every color class is spanning and has diameter at most four.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.04900
 Bibcode:
 2021arXiv210904900C
 Keywords:

 Mathematics  Combinatorics;
 05C55
 EPrint:
 4 pages