Induced subgraphs density. IV. New graphs with the Erdős-Hajnal property

Abstract

Erdős and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$). Alon, Pach, and Solymosi reduced the Erdős-Hajnal conjecture to the case when $H$ is {\em prime} (that is, $H$ cannot be obtained by vertex-substitution from smaller graphs); but until now, it was not shown for any prime graph with more than five vertices. We will provide infinitely many prime graphs that satisfy the conjecture. Let $H$ be a graph with the property that for every prime induced subgraph $G'$ with $|G'|\ge 3$, $G'$ has a vertex of degree one and a vertex of degree $|G'|-2$. We will prove that every graph $H$ with this property satisfies the Erdős-Hajnal conjecture, and infinitely many graphs with this property are prime. More generally, say a graph is {\em buildable} if every prime induced subgraph with at least three vertices has a vertex of degree one. We prove that if $H_1$ and $\overline{H_2}$ are buildable, there exists $c>0$ such that every graph $G$ that is both $H_1$-free and $H_2$-free has a clique or a stable set of size at least $|G|^c$. Our proof uses a new technique of ``iterative sparsification'', where we pass to a sequence of successively more restricted induced subgraphs. This approach also extends to ordered graphs and to tournaments. For ordered graphs, we obtain a theorem which significantly extends a recent result of Pach and Tomon about excluding monotone paths; and for tournaments, we obtain infinitely many new prime tournaments that satisfy the Erdős-Hajnal conjecture (in tournament form).

Figures (5)

• Figure 1: The two six-vertex prime graphs in $\mathcal{H}$, and one on seven vertices.
• Figure 2: Start with a path ($a_2\hbox{-} b_3\hbox{-} a_6 \hbox{-} b_7\hbox{-} a_{10}\hbox{-} b_{11}$ in this case), add a leaf at every vertex, add an isolated vertex $b_1$, and take a bipartition $(A,B)$, numbered as shown. Now make $A$ a clique; and make $a_i, b_j$ adjacent if $i\ge j+4$.
• Figure 3: The six-vertex graphs not containing $P_5$ or $\overline{P_5}$ that remain open.
• Figure 4: With $H$ as shown, $H\in \mathcal{J}$, and so $\{H,\overline{H}\}$ is viral.
• Figure 5: The four-vertex prime ordered graphs (up to complements and reversal).

Theorems & Definitions (39)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.7
• Conjecture 1.8
• Theorem 1.9
• Theorem 1.10