Graph Theory Questions Medium
The chromatic polynomial of a graph is a polynomial that counts the number of ways to color the vertices of the graph using a given number of colors, such that no adjacent vertices have the same color. It is denoted by P(G, λ), where G represents the graph and λ represents the number of colors.
The chromatic polynomial is defined recursively as follows:
1. If the graph G has no edges, then P(G, λ) = λ^n, where n is the number of vertices in G. This means that each vertex can be colored with any of the λ colors independently.
2. If the graph G has an edge between vertices u and v, then P(G, λ) = P(G - uv, λ) - P(G - uv, λ-1). This means that we consider two cases: coloring u and v with the same color (P(G - uv, λ-1)) and coloring them with different colors (P(G - uv, λ)). We subtract the former from the latter to avoid overcounting.
By applying these recursive rules, we can calculate the chromatic polynomial of a graph. The coefficients of the polynomial provide information about the number of colorings for each value of λ. The chromatic polynomial is a useful tool in graph theory for studying graph coloring problems and determining properties of graphs based on their coloring behavior.