Clique number

Definition

The clique number of a graph is defined as the size of the largest complete subgraph within that graph, meaning it is the maximum number of vertices in a clique. This concept connects to various important ideas such as vertex coloring, where the clique number helps determine the chromatic number, and it plays a critical role in understanding the structure of maximum cliques, independent sets, and their relationships with vertex covers. It also relates to extremal graph theory through Turán's theorem, which provides insights into how cliques interact with the overall structure of a graph.

5 Must Know Facts For Your Next Test

1. The clique number is often denoted by $$ ext{cl}(G)$$ for a graph $$G$$, and it represents the largest size of any complete subgraph contained within $$G$$.
2. In graphs with a higher clique number, it's generally more challenging to find proper vertex colorings since larger cliques require more colors to avoid adjacent vertices sharing the same color.
3. The relationship between independent sets and cliques indicates that if a graph has a large clique number, it typically has a smaller independent set size, showcasing their inverse relationship.
4. Turán's theorem states that for any graph with $$n$$ vertices and no cliques larger than size $$r$$, the maximum number of edges it can have is related to $$n$$ and $$r$$, emphasizing how cliques influence edge density.
5. Cliques can be used to model real-world situations like social networks where connections among individuals represent friendships; thus, finding the clique number can help identify tightly-knit groups.

Review Questions

• How does the concept of clique number impact the determination of a graph's chromatic number?
  • The clique number plays a significant role in determining a graph's chromatic number because larger cliques imply that more colors are needed to ensure that no two adjacent vertices share the same color. Specifically, if a graph contains a complete subgraph with $$k$$ vertices (a clique of size $$k$$), then at least $$k$$ colors are necessary for proper vertex coloring. Thus, the chromatic number is always at least as large as the clique number.
• Discuss how Turán's theorem relates to the concept of clique number and its implications on edge counts in graphs.
  • Turán's theorem provides a framework for understanding how cliques limit the edge count in graphs. The theorem states that for any graph containing no complete subgraphs larger than size $$r$$, there is a maximum edge count determined by the number of vertices. This directly links the idea of clique number with extremal properties of graphs, indicating that higher clique numbers constrain potential edge density, leading to important insights in both theoretical and applied contexts.
• Evaluate the relationship between maximum cliques and independent sets in terms of their sizes and implications for graph properties.
  • The relationship between maximum cliques and independent sets illustrates an interesting duality in graph theory. As the size of a maximum clique increases, the size of the largest independent set tends to decrease because more vertices are connected within that clique. This connection can be essential when analyzing graph properties such as stability and coloring strategies since understanding one aspect often leads to insights about the other. Ultimately, this interplay is vital for solving optimization problems related to network design and resource allocation.

Related terms

Complete Graph: A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge, representing a clique of size equal to the number of vertices.

Chromatic Number: The chromatic number of a graph is the minimum number of colors needed to color its vertices so that no two adjacent vertices share the same color; it is closely related to the clique number.

Independent Set: An independent set in a graph is a set of vertices no two of which are adjacent; the size of the largest independent set can be influenced by the clique number.