Extremal Graphs for n ≥ 11: Theory & Constructions

• Extremal graphs for n ≥ 11 are defined by achieving tight edge bounds under forbidden minors, notably with exₘ(n, P) = 5n–9 for Petersen-minor-free graphs.
• The (K₉,2)-cockade constructions uniquely realize these bounds by gluing K₉ components without forming full 3-connectivity, highlighting structural subtleties.
• Analyses of path-intersection separators and coloring/arboricity implications reveal practical insights and open directions for extending classification beyond n = 11.

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 (Hendrey et al., 2015).

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 $n \equiv 2 \pmod{7}$ (Hendrey et al., 2015).

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 (Hendrey et al., 2015).

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 (Dowden, 2015).
• 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 $y$ (Gutiérrez et al., 2021).

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 (Hendrey et al., 2015).
• 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 (Hendrey et al., 2015, Dowden, 2015, Gutiérrez et al., 2021). 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.