 Chromatic polynomial

For each positive integer $t$, the chromatic polynomial $P_G(t)$ counts the proper vertex colourings of a finite graph $G$ using a fixed palette of $t$ colours.

The **chromatic polynomial** is a polynomial that encodes the number of ways to color the vertices of a graph using a given number of colors such that no two adjacent vertices share the same color. Specifically, for a graph \( G \), the chromatic polynomial \( P(G, k) \) is defined as the number of ways to color the graph with \( k \) colors. ### Key Properties 1.