3.4 Saturation Problems in Graphs

Saturation in extremal graph theory explores the minimum number of edges needed to guarantee specific subgraphs. It's all about finding that sweet spot where adding just one more edge creates the subgraph you're looking for.

This concept helps us understand graph structure and design efficient algorithms. By studying saturation numbers for various subgraphs like cliques and paths, we gain insights into how graphs evolve and when certain patterns emerge.

Saturation in Extremal Graph Theory

Saturation in extremal graph theory

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• Concept deals with minimum number of edges needed to guarantee presence of specific subgraph in graph
  • Graph $G$ is $H$-saturated if does not contain $H$ as subgraph, but adding any new edge to $G$ creates copy of $H$
• Saturation number $\text{sat}(n, H)$ is minimum number of edges in $n$-vertex $H$-saturated graph
  • Determining saturation number helps understand structure of graphs on verge of containing specific subgraph
• Saturation problems significant because provide insights into minimum conditions for emergence of specific substructures in graphs
  • Help design efficient algorithms for detecting subgraphs (cliques, paths)
  • Understand trade-offs between number of edges and presence of specific subgraphs (cycles, stars)

Determination of saturation numbers

• Saturation number for complete graph $K_t$:
  • $\text{sat}(n, K_t) = (t-2)(n-t+2) + \binom{t-2}{2}$ for $n \geq t \geq 3$
  • Represents minimum number of edges to ensure any additional edge creates $K_t$ subgraph in $n$-vertex graph
• Saturation number for path $P_t$ on $t$ vertices:
  • $\text{sat}(n, P_t) = n - t + \left\lfloor \frac{t}{2} \right\rfloor + 1$ for $n \geq t \geq 2$
  • Gives minimum number of edges needed to guarantee adding any new edge to $n$-vertex graph creates path of length $t$
• Determining saturation number for other specific graphs and subgraphs involves analyzing their structure and conditions for their emergence
  • Cycles
  • Stars
  • Cliques

Saturation vs Turán-type problems

• Turán-type problems focus on determining maximum number of edges in graph that does not contain specific subgraph
  • Turán number $\text{ex}(n, H)$ represents maximum number of edges in $n$-vertex graph that does not contain $H$ as subgraph
• Saturation problems can be seen as dual of Turán-type problems
  • Turán-type problems aim to maximize number of edges while avoiding specific subgraph
  • Saturation problems focus on minimizing number of edges while being on verge of creating that subgraph
• Relationship between saturation and Turán numbers:
  • $\text{sat}(n, H) \leq \text{ex}(n, H) + 1$
  • Inequality holds because any $H$-saturated graph with minimum number of edges can be obtained by adding one edge to graph with maximum number of edges that does not contain $H$

Minimum edges for subgraph presence

• To solve saturation problems, consider structure of specific subgraph and conditions for its emergence
• Constructive approaches involve designing graphs that are $H$-saturated and have minimum number of edges
  • Provide upper bounds on saturation number
• Lower bound techniques prove any graph with fewer edges than claimed saturation number cannot be $H$-saturated
  • Use pigeonhole principle, counting arguments, and graph theory techniques to establish lower bounds
• Solving saturation problems may involve:
  1. Case analysis
  2. Considering different ranges of number of vertices
  3. Analyzing structure of subgraph in question
• Different strategies may be required for different types of subgraphs
  • Paths
  • Cycles
  • Cliques
  • Stars

Applying Saturation Concepts

Minimum edges for subgraph presence

• Example problem: Determine minimum number of edges required in graph with 10 vertices to ensure any additional edge creates triangle ($K_3$)
  • Solution:
    1. Use formula for saturation number of complete graph: $\text{sat}(n, K_t) = (t-2)(n-t+2) + \binom{t-2}{2}$
    2. Substitute $n=10$ and $t=3$:
    • $\text{sat}(10, K_3) = (3-2)(10-3+2) + \binom{3-2}{2} = 1 \cdot 9 + 0 = 9$
    1. Graph with 10 vertices must have at least 9 edges to ensure any additional edge creates triangle
• Example problem: Prove any graph with $n$ vertices and $\text{sat}(n, P_4) + 1$ edges must contain path of length 4
  • Solution:
    1. By definition of saturation, any graph with $\text{sat}(n, P_4)$ edges is $P_4$-saturated, meaning it does not contain path of length 4, but adding any new edge creates such path
    2. If graph has $\text{sat}(n, P_4) + 1$ edges, it has one more edge than minimum required to be $P_4$-saturated
    3. Additional edge must create path of length 4 in graph, otherwise graph would be $P_4$-saturated with fewer than $\text{sat}(n, P_4)$ edges, contradicting definition of saturation number
    4. Therefore, any graph with $n$ vertices and $\text{sat}(n, P_4) + 1$ edges must contain path of length 4