# Find the chromatic polynomial of a graph

My answer: $p(g,k) = k(k-1)^4(k-2)(k-3)$

I'm new to this subject so was hoping if one of you could check my answer. Thanks.

Vertices:

• Your answer says $p(g,3)=0$ so there are no ways to color it with $3$ colors? But I just did it: I colored the top vertex red, its two neighbors white, the the four bottom vertices alternately red and blue. So I think your answer must be wrong.
– bof
Commented Jun 17, 2016 at 9:45
• Do you know about Decomposition theorem in graph coloring? Commented Jun 17, 2016 at 9:53
• @mathlover Nope. Does it have any relevance to finding the chromatic polynomial of a graph? Commented Jun 17, 2016 at 10:39
• Yes..See any standard textbook of graph theory under section graph vertex coloring. Commented Jun 17, 2016 at 10:50
• @bof So it would be chromatic polynomial = $k^3(k-1)^2(k-3)^2$? or do i plus them together not times? Commented Jun 17, 2016 at 13:05

By Maple software, the chromatic polynomial of your graph, $G$, is $P(G,x)=x(x-1)(x^2-2x+2)(x-2)^3$. Its chromatic number is $3$ with vertex coloring $[[2, 5], [1, 3, 6], [4, 7]]$.