Graph Limit Seminar
A solution to the 2/3 conjecture
ETH Zürich 2015, Seminar
Paper: A solution to the 2/3 conjecture, Rahil Baber and John Talbot
Definition 1.
• A complete graph is a graph G = (V, E) such that E = {(u, v) : u 6= y ∈ G}.
• An r-coloring of the edges of a graph G is a map from E(G) → {1, . . . , r}.
• Let G be an r-colored graph. Let A, B ⊂ V (G), c ∈ {1, . . . , r}. Then we say A cdominates B if ∀b ∈ B\A : ∃a ∈ A such that (a, b) is colored c. We say A strictly
c-dominate B if ∀b ∈ B this holds.
• Let v ∈ V . Then we denote with Av the set of colors of the edges incident to v.
• Let Gn be a sequence of graph constructed from a complete 3-colored graph Ĝ as follows.
Gn is a blowup of Ĝ by substitute each vertex from Ĝ with n vertices. The coloring of
the edges will be:
1. edges in between the n vertices of a vertex v ∈ Ĝ are colored uniformly at random
with colors from Av , and
2. edges between different n-vertex sets of vertices u, v ∈ Ĝ are colored in the same
way as the edge (u, v) in Ĝ.
• Two 3-colored graphs are isomorphic if they can be made identical by permutating their
vertices.
Previous Results
Theorem 2 (Erdős et al.). For any t ∈ N and any 2-colored complete graph on n vertices,
there exists a color c and a set of at most t vertices that c-dominate at least (1 − 21t )n of the
vertices.
Conjecture 3 (Erdős et al.). For any 3-colored complete graph there exists a color c and a
set of at most 3 vertices that c-dominates at least 2/3 of the vertices.
Their result
Theorem 4. For any 3-coloring of the edges of a complete graph on n ≥ 3 vertices, there
exists a color c and a set of 3 vertices that strongly c-dominates at least 2n/3 vertices.
Proof. Let us assume that is not the case, i.e. let Ĝ be a 3-colored complete graph on k ≥ 3
vertices such that every set of 3 vertices strongly c-dominates strictly less than 2k/3 vertices
for each color c.
Lemma 5. There exists a vertex v ∈ V (Ĝ) with |Av | = 3.
The contradiction to our assumption will follow from the density of X (see Fig. 1) in Gn .
Corollary 6. With probability 1 − o(1) we have dX (Gn ) ≥ k −|V (X)| · 3−|E(X)| + o(1), where
o(1) → 0 if n → ∞.
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Figure 1: The 3-colored graph X.
Definition 7.
• For a 3-colored complete graph and a color c we call a set of 3 vertices
{x, y, z} a good set for c if either
1. at least two of the edges (x, y), (x, z), (y, z) are colored c, or
2. one of the edges, wlog. (x, y) is colored c and the remaining vertex z satisfies
|Az ∪ {c}| = 3.
• A σ-flag F , where |σ| = 3, is c-good if the coloring of F imply that σ is a good set for
c in F .
Lemma 8. Any good set for c in Gn strongly c-dominates at most 2/3 + o(1) of the vertices
with probability 1 − o(1).
By encoding this lemma using flag algebras we get then ae contradiction to Corollary 6:
Lemma 9. With probability 1 − o(1) we have dX (Gn ) = o(1)
Proof-idea: We first find an upper bound of dX (Gn ) by some functions αH , βH in the form
of
dX (Gn ) ≤ max (dX (H) + αH (Q) + βH (µ)) + o(1)
H∈H6
where αH depends on a set of positive semi-definite matrices Q (not fixed) and βH depends
on a non-negative function µ. To proof then the result, one can use semidefinite programming
to find a good pair Q, µ such that maxH∈Hl (dX (H) + αH (Q) + βH (µ)) = 0.
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