A new bound for the 2/3 conjecture

Daniel Král' (1) , Chun-Hung Liu (2) , Jean-Sébastien Sereni (3) , Peter Whalen (2) , Zelealem Yilma (4)

1 DIMAP - Centre for Discrete Mathematics and its Applications [Warwick]
2 School of Mathematics - Georgia Institute of Technology
3 ORPAILLEUR - Knowledge representation, reasonning
4 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Mots clés: Monochromatic domination; colored complete graph; flag algebra

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Daniel Král', Chun-Hung Liu, Jean-Sébastien Sereni, Peter Whalen, Zelealem Yilma. A new bound for the 2/3 conjecture. [Research Report] Loria & Inria Grand Est. 2012. ⟨hal-00686989v2⟩