---
title: 'Extremal Graphs for n ≥ 11: Theory & Constructions'
url: https://www.emergentmind.com/topics/extremal-graphs-for-n-ge-11
type: topic
---

# Extremal Graphs for n ≥ 11: Theory & Constructions

Extremal graphs for $n \geq 11$ pertain to the structure and parameters of graphs that achieve or nearly achieve maximal or minimal values of certain invariants under prescribed forbidden configurations or substructures. Key extremal phenomena for $n \geq 11$ include tight results for subgraph exclusion (e.g., forbidden minors or cycles), sharp characterizations of extremal graphs, and the first appearance or absence of certain subtle combinatorial properties.

## 1. Extremal Functions for Forbidden Minors

For the Petersen graph $P$, Hendrey and Wood determined the extremal function $\operatorname{ex}_m(n, P)$, the maximum number of edges in a graph with $n$ vertices that excludes $P$ as a minor. The fundamental result is:
- Every $n$-vertex graph with $|E(G)| \geq 5n-8$ contains the Petersen graph as a minor.
- The extremal function is $\operatorname{ex}_m(n, P) = 5n-9$ for all $n \geq 3$.
- Equality $\operatorname{ex}_m(n, P) = 5n-9$ is achieved if and only if $n \equiv 2 \pmod{7}$; these extremal graphs are uniquely characterized as $(K_9,2)$-cockades [1508.04541].

This result is best possible in both the edge bound and structural sense for $n \geq 11$.

## 2. $(K_9,2)$-Cockades: Extremal Constructions

A $(K_9,2)$-cockade is constructed by repeatedly gluing copies of the complete graph $K_9$ along shared $K_2$ subgraphs, without creating 3-connectivity:
- The initial case is $K_9$ itself.
- When two such graphs are joined on a shared edge ($K_2$), the edges in the overlap are counted only once.
- Every $(K_9,2)$-cockade on $n$ vertices has exactly $5n-9$ edges and contains no Petersen minor because the resulting structure is never 3-connected, in contrast to the Petersen graph.
- Cockade constructions provide all extremal examples for $\operatorname{ex}_m(n, P)$ with $n \geq 11$ and $n \equiv 2 \pmod{7}$ [1508.04541].

## 3. Near-Extremal Characterization and Edge Deficits

For $n \geq 11$, graphs with edge counts in $5n-11 \leq |E(G)| \leq 5n-9$ are nearly extremal regarding Petersen minors:
- If $|E(G)| = 5n-10$, $G$ arises by deleting exactly one edge from a $(K_9,2)$-cockade.
- If $|E(G)| = 5n-11$, $G$ arises by deleting exactly two edges from a $(K_9,2)$-cockade.
- Any deviation from this structure guarantees the existence of a Petersen minor; hence, the classification for near-extremal graphs is exhaustive within these parameters [1508.04541].

## 4. Extremal Results for Planar Graphs with Forbidden Small Cycles

For planar graphs on $n \geq 11$ vertices excluding small cycles:
- The extremal function for $C_5$-free planar graphs is $\operatorname{ex}_\mathcal{P}(n, C_5) \leq (12n-33)/5$.
- The bound is tight for infinitely many $n$; explicit constructions exist achieving equality [1512.04385].
- The extremal family is built from recursive and gadget-based planar constructions (using special triangulations, “diamond-holder”, and “snowflake” components).
- For $C_4$-free planar graphs, $\operatorname{ex}_\mathcal{P}(n, C_4) \leq \frac{15}{7}(n-2)$ for $n \geq 4$, again with infinite families realizing equality and a simpler structure than the $C_5$-free case.

A summary of tight extremal edge functions for planar graphs:

| Forbidden Subgraph | Bound on $|E(G)|$ (for $n \geq 11$)        | Tight for infinitely many $n$?  |
|---------------------|----------------------------------------------|----------------------------------|
| $C_4$               | $\frac{15}{7}(n-2)$                         | Yes                             |
| $C_5$               | $\frac{12n-33}{5}$                          | Yes                             |

These results establish sharp upper bounds and constructions.

## 5. Path Intersection Separators at the Threshold $n=11$

In the context of path intersections, a minimality threshold occurs at $n=11$ for the property that intersections of two longest paths are always separators:
- For every connected simple graph $G$ with $n \leq 10$, and any two longest paths $P,Q$, $V(P)\cap V(Q)$ is a separator.
- For $n=11$, there exists a unique minimal example $G_{11}$ where two longest paths have a non-separating intersection. $G_{11}$ consists of a spine of 9 vertices $\{v_1,\dots,v_9\}$, with two pendant vertices $x$ and $y$ attached to $v_1, v_2$ and $v_8, v_9$, respectively, and a single edge joining $x$ and $y$ [2105.11633].

For $n > 11$, additional counterexamples can be constructed by attaching pendant trees or blocks to the $G_{11}$ core, but a full structural classification remains open.

## 6. Colouring and Arboricity Implications

Sharp corollaries follow for minor-exclusion extremal graphs:
- Any Petersen-minor-free graph is 9-colourable; with $|P|=10\leq 2 \times 5$, 8 colours suffice for $n > 10$.
- All such graphs have maximum average degree less than 10 and are thus 9-degenerate.
- Vertex arboricity is at most 5; every such graph can be partitioned into 5 forests [1508.04541].
- These bounds are tight, realized by the $(K_9,2)$-cockades.

| Property                   | Bound                       | Extremal Example Achieving Bound           |
|----------------------------|-----------------------------|--------------------------------------------|
| Chromatic number           | $\leq 9$ (or 8 for $n>10$)  | $(K_9,2)$-cockades ($\chi=9$)              |
| Vertex-arboricity          | $\leq 5$                    | $(K_9,2)$-cockades (arboricity $=5$)       |

These corollaries are direct consequences of the underlying edge-density results and are best possible.

## 7. Summary and Open Directions

For $n \geq 11$, extremal graph theory provides the following:
- Sharp extremal numbers for forbidden minors (notably, $\operatorname{ex}_m(n,P) = 5n-9$ for Petersen-minor-free graphs, with unique cockade structures when equality holds).
- Explicit extremal constructions for planar graphs excluding small cycles, with tight analytic and combinatorial proofs.
- Structural thresholds such as the minimal $n=11$ for the failure of path-intersection separator properties, and explicit minimal non-separating examples.
- Tight and optimal chromatic and arboricity bounds for extremal classes.

Current research for $n > 11$ considers the full classification of graphs with forbidden substructure near the extremal edge bounds, especially regarding path-intersection properties beyond minimal examples [1508.04541, 1512.04385, 2105.11633]. A plausible implication is that further block-attaching techniques may yield the complete landscape beyond $n=11$ for certain separator problems, while for extremal density for forbidden minors and cycles, the structure is fully characterized for large $n$ by these described constructions.

Source: https://www.emergentmind.com/topics/extremal-graphs-for-n-ge-11