Home >> Science & Nature >> Science

How to Determine a Chromatic Number From a Polynomial

A chromatic number is used in graph theory to show the number of colors needed to color in the vertices of a graph, i.e., the points of intersection, without any adjacent vertices having the same color. For example, a triangle would have a chromatic number of three, but a square would have a chromatic number of two. A chromatic polynomial is a similar concept in graph theory, but it seeks the most number of ways a graph can be colored using a certain number of colors. Chromatic polynomials are known for only certain types of graphs.

Instructions

- 1
  Figure out the chromatic polynomial for a triangle graph with the following formula: t((t ͨ2; 1)^2)(t ͨ2; 2), where "t" is the number of colors to use. A triangle graph shows a shape made of many K to the 2rd power of triangles. Simply plug in the number of colors you want the graph to have into the formula to find the chromatic polynomial. For example, for five colors, the chromatic number is: 5((5-1)^2)(5-2), which is: 240.
- 2
  Find the chromatic polynomial for a Complete Graph, which is a shape that has every pair of distinct vertices connected by an edge. Use this formula: t(t-1)(t-2) on up to t-n, where "n" is the number of edges of the graph and "t" is the number of colors to graph the vertices. For a complete graph with two edges and four colors, the chromatic polynomial is: 4(4-1)(4-2)=24.
- 3
  Calculate the chromatic polynomial for a tree graph with the formula:
  
  t(t ͨ2; 1)^(n ͨ2; 1)
  
  A tree graph is made up of nodes or vertices that branch off one another the way tree branches do. In this formula, "n" is the number of vertices of the tree. So a tree graph with five vertices and two colors would have a chromatic polynomial of: 2(2-1)^(5-1)=16.
- 4
  Calculate the chromatic polynomial for a Cycle Graph, which displays a number of vertices connected in a ring shape. Use this formula:
  
  (t ͨ2; 1)^n + ( ͨ2; 1)^(n)(t ͨ2; 1)
  
  In this formula, "n" is the number of vertices and "t" is the number of colors. A Cycle Graph with two vertices and two colors has a chromatic polynomial of: (2-1)^2+(-1)^2)(2-1)=2.
- 5
  Calculate the last kind of graph for which the formula of the chromatic polynomial is known, the Peterson Graph, with the following, forbidding formula:
  
  t(t ͨ2; 1)(t ͨ2; 2)(t7 ͨ2; 12t6 + 67t5 ͨ2; 230t4 + 529t3 ͨ2; 814t2 + 775t ͨ2; 352)
  
  A Peterson Graph is a graph with 10 vertices and 15 edges. In this formula, "t" is the number of colors to use for the graph. So a chromatic polynomial with two colors for a Peterson Graph -- 2(2 ͨ2; 1)(2 ͨ2; 2)(2*7 ͨ2; 12*2*6 + 67*2*5 ͨ2; 230*2*4 + 529*2*3 ͨ2; 814*2*2 + 775*2 ͨ2; 352) -- is 0, because the first part of the equation equals zero and cancels the second part. This makes sense because a chromatic polynomial expresses the number of colors needed so that no two adjacent vertices have the same color. This doesn't work in the Peterson Graph because vertices are paired next to each other.

Science