In graph theory, "k" can represent several different things depending on the context. Here are some common meanings:
1. K-Clique:
- Definition: A k-clique is a complete subgraph with k vertices, where every pair of vertices is connected by an edge.
- Example: A 3-clique is a triangle, a 4-clique is a tetrahedron, and so on.
2. K-Coloring:
- Definition: A k-coloring of a graph assigns one of k colors to each vertex such that no two adjacent vertices have the same color.
- Example: A graph is 3-colorable if it can be colored with at most three colors.
3. K-Regular Graph:
- Definition: A k-regular graph is a graph where every vertex has k neighbors.
- Example: A 3-regular graph is also known as a cubic graph.
4. K-Path:
- Definition: A k-path is a simple path with k edges.
- Example: A 3-path is a path with three edges and four vertices.
5. K-Connected Graph:
- Definition: A k-connected graph is a graph where at least k vertices need to be removed to disconnect the graph.
- Example: A 2-connected graph is also known as a biconnected graph.
6. K-Factor:
- Definition: A k-factor of a graph is a spanning subgraph where every vertex has degree k.
- Example: A 1-factor is a perfect matching.
7. K-Chromatic Number:
- Definition: The k-chromatic number of a graph is the minimum number of colors needed to color the graph such that no two adjacent vertices have the same color.
- Example: The chromatic number of a complete graph with k vertices is k.
These are some of the common meanings of "k" in graph theory. The specific meaning of "k" will depend on the context and the specific graph theory concept being discussed.
