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Every graph on at most three vertices has the Erdős–Hajnal property
Statement
Every finite graph with has the Erdős–Hajnal property.
Facts & Assumptions
Given: A finite graph with at most three vertices.
A graph has the Erdős–Hajnal property when its hereditary -free class has a positive exponent (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class, -free and -free graphs under the induced-subgraph convention, Every class defined by forbidden induced subgraphs is hereditary).
For every , the class of -free graphs has the Erdős–Hajnal property (For every , the class of -free graphs has the Erdős–Hajnal property).
Every -free graph satisfies , so has the property (Every -free graph satisfies ).
A graph and its complement have exactly the same Erdős–Hajnal constants (A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants).
The graphs and have the standard edge sets, and is the null graph (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices); complementation replaces the edge set by all missing pairs (Graph isomorphisms, automorphisms and graph complements).
Proof
[assume-case null] If , every graph contains the unique empty induced embedding of , so the -free class has no members and [L1] is vacuously satisfied by every positive exponent.
[assume-case nonnull] Suppose . Up to isomorphism and complementation, is one of , or : this follows by the edge count for orders at most two, and for order three by separating the cases of zero, one, two, or three edges.
Each complete case has the property by [L2], the path case has it by [L3], and every complementary case has it by [L4].
The cases are exhaustive, so every graph on at most three vertices has the Erdős–Hajnal property.
Depends on
- The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class
- $H$-free and $\mathcal F$-free graphs under the induced-subgraph convention
- For every $t\ge1$, the class of $K_t$-free graphs has the Erdős–Hajnal property
- Every $P_3$-free graph $G$ satisfies $\operatorname{hom}(G)\ge\sqrt{|V(G)|}$
- A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants
- Every class defined by forbidden induced subgraphs is hereditary
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
- Graph isomorphisms, automorphisms and graph complements
Used by
- The graph H₀ has the Erdős-Hajnal property Corollary
- The polynomial Rödl witness need not be the whole graph Counterexample
- Substituting an edge for an endpoint of P₃ gives a four-vertex graph with the Erdős–Hajnal property Example
- The singleton family {P₃} is viral Example
- Deleting a leaf and a co-leaf preserves virality of a finite forbidden family Theorem
- Every graph on at most four vertices has the Erdős-Hajnal property Theorem
Dependency tree · two levels
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Sources
- M. Chudnovsky, The Erdos-Hajnal Conjecture: A Survey, sec. 2 (standard reference, not scraped)