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The two six-vertex prime -graphs have the Erdős-Hajnal property
Statement
Both the left and the right six-vertex prime -graphs have the Erdős-Hajnal property.
Facts & Assumptions
Given: The left and right six-vertex prime -graphs.
Every graph on at most four vertices has the Erdős-Hajnal property (Every graph on at most four vertices has the Erdős-Hajnal property).
For a single graph, the Erdős-Hajnal property is equivalent to virality (For a single graph, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent).
Deleting a leaf and a co-leaf from two forbidden graphs preserves virality (Deleting a leaf and a co-leaf preserves virality of a finite forbidden family).
A graph and its complement have the same Erdős-Hajnal constants (A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants).
Substituting one Erdős-Hajnal graph for a vertex of another preserves the Erdős-Hajnal property (Alon–Pach–Solymosi: if and have the Erdős–Hajnal property, so does the graph obtained from by substituting for a vertex).
In the graph of The left six-vertex prime -graph, is a leaf and is a co-leaf: is adjacent to and nonadjacent to . The graph is the substitution of the edge for the second vertex of the path . The graph is the substitution of the path for one vertex of a two-vertex stable set whose other vertex is .
The right six-vertex prime -graph is the complement of the left one by definition.
Proof
The two factor graphs in each substitution of [F1] have at most four vertices. By [L1] and [L5], both and have the Erdős-Hajnal property. By [L2], the singleton families and are viral.
Apply [L3] to the family , using in the leaf-deletion slot and in the co-leaf-deletion slot. Step 1.1 verifies both viral hypotheses, so is viral. The reverse direction of [L2] gives the Erdős-Hajnal property for .
Let be the right six-vertex prime -graph. By [F2], we have , so [L4] transfers the Erdős-Hajnal property from to .
Therefore both six-vertex prime -graphs have the Erdős-Hajnal property.
Depends on
- The left six-vertex prime $\mathcal H$-graph
- The right six-vertex prime $\mathcal H$-graph
- Every graph on at most four vertices has the Erdős-Hajnal property
- Alon–Pach–Solymosi: if $H_1$ and $H_2$ have the Erdős–Hajnal property, so does the graph obtained from $H_1$ by substituting $H_2$ for a vertex
- Deleting a leaf and a co-leaf preserves virality of a finite forbidden family
- For a single graph, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent
- A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants
Used by
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Sources
- Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. IV. New graphs with the Erdős-Hajnal property, Theorem 1.4 (standard reference, not scraped)
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 2 discussion (standard reference, not scraped)