The text discusses the stability of graph properties in the context of graph theory, particularly focusing on Hamiltonian cycles. Let's break it down step by step for clarity:

1. Key Definitions

• Property $P$:

  • A property defined on graphs of order $n$, meaning graphs with $n$ vertices. For example, the property $P$ could be "the graph is Hamiltonian" (i.e., it contains a Hamiltonian cycle).

• $k$-stability of a property:

  • A property $P$ is $k$-stable if the following holds:
    • If a graph $G + uv$ (where $uv$ is an edge added to $G$) has property $P$,
    • And the sum of the degrees of $u$ and $v$ in $G$, denoted $d_G(u) + d_G(v)$, is at least $k$,
    • Then $G$ itself must also have property $P$.

This concept measures the "resilience" of a property to changes like adding edges.

2. Statement: The Hamiltonian Cycle Property is $n$-Stable

• A Hamiltonian cycle is a cycle that visits every vertex of a graph exactly once and returns to the starting vertex.

The claim here is:

• The property of containing a Hamiltonian cycle is $n$-stable, where $n$ is the number of vertices in the graph.

This means:

• If adding an edge $uv$ to $G$ makes $G + uv$ Hamiltonian,
• And if $d_G(u) + d_G(v) \geq n$,
• Then the original graph $G$ itself must already contain a Hamiltonian cycle.

3. Proof Outline

The proof proceeds as follows:

1. Assumption:

   • Suppose $G + uv$ is Hamiltonian, but $G$ is not.

2. Structure of $G + uv$:

   • Since $G + uv$ is Hamiltonian, it contains a Hamiltonian path of the form: $u = u_1, u_2, \ldots, u_n = v.$
   • The vertices $u$ and $v$ are connected by the edge $uv$.

3. Key Observation:

   • The condition $d_G(u) + d_G(v) \geq n$ ensures that there is a vertex $u_i$ such that:
     • $u$ is adjacent to $u_{i+1}$, and
     • $v$ is adjacent to $u_i$.

4. Conclusion:

   • Using this structure, you can form a Hamiltonian cycle in $G$ by connecting: $u_1, u_{i+1}, u_{i+2}, \ldots, u_n, u_i, u_{i-1}, \ldots, u_1.$
   • Thus, $G$ itself must already contain a Hamiltonian cycle, contradicting the assumption that $G$ is not Hamiltonian.

4. Implications and Similar Results

The text mentions that similar arguments can be applied to prove related stability results. For instance:

• $P$ could represent other graph properties (like connectivity or specific subgraph containment).

Key Takeaway:

The $n$-stability of the Hamiltonian cycle property is a fundamental result showing that certain conditions on vertex degrees ensure the property persists even without adding edges.

Would you like me to delve into any of the specific steps, such as why the degree condition is crucial? Let me know!

Related Questions:

1. What is the significance of $d_G(u) + d_G(v) \geq k$ in the context of graph stability?
2. How does the concept of $k$-stability relate to other properties like connectivity?
3. Can $k$-stability be applied to properties like bipartiteness or Eulerian circuits?
4. What are examples of graphs where the Hamiltonian cycle property is not $n$-stable?
5. How do Ore's theorem and Dirac's theorem relate to this discussion?

Tip:

When analyzing stability in graphs, always consider how adding edges impacts the degrees of vertices and how those changes might affect the global property being studied.