problem to determine the chromatic polynomial of a graph

Question

for a homework graph theory, I'm asked to determine the chromatic polynomial of the following graph

enter image description here

For the Descomposition Theorem of Chromatic Polynomials. if G=(V,E), is a connected graph and e belong E

P (G, λ) = P (Ge, λ) -P(Ge', λ)

where Ge denotes de subgraph obtained by deleting de edge e from G (Ge= G-e) and Ge' is the subgraph obtained by identifying the vertices {a,b} = e

When calculating chromatic Polynomials, i shall place brackets about a graph to indicate its chromatic polynomial. removes an edge any of the original graph to calculate the chromatic polynomial by the method of decomposition.

enter image description here

 P (G, λ) = P (Ge, λ)-P (Ge', λ) = λ (λ-1)^4 - [λ(λ-1)*(λ^2 - 3λ + 3)]

But the response from the answer key and the teacher is:

P (G, λ) = λ (λ-1)(λ-2)(λ^2-2λ-2)

I have operated on the polynomial but I can not reach the solution that I ask .. what am I doing wrong?

Interesting problem but i think you might do better with getting an answer at: http://cstheory.stackexchange.com/ or http://math.stackexchange.com/ — Paul Sasik, Apr 20 '11 at 00:14
yeah but in both pages i can't post any images because im new user. — franvergara66, Apr 20 '11 at 00:17
Go ahead and open new questions on both sites and i'll edit the questions for you with the images as they are here. Reply here, of course, so I know when the Qs are ready. — Paul Sasik, Apr 20 '11 at 00:31
http://math.stackexchange.com/questions/33946/problem-to-determine-the-chromatic-polynomial-of-a-graph — franvergara66, Apr 20 '11 at 01:54
Sounds like you got a great answer. And a good idea to link back to SO for the original question with images. So you're good to go? — Paul Sasik, Apr 20 '11 at 02:13
the question was not solved, but it gave me an idea of how to do it, I think I'll close the question — franvergara66, Apr 20 '11 at 21:25

score 3 · Answer 1 · answered Mar 12 '13 at 23:29

Your answer is correct, and so is the teacher's --they are equal. [By the way, nice picture and explanation.]

An odd-cycle can have no 2-coloring, hence the 5-cycle can have no 2-coloring, so its chromatic polynomial, f(x), must have x * [x - 1] * [x - 2]

as a divisor. If you combine your expression for f(x) and divide out the

x * [x - 1]

then you'll find that what remains is divisible by [x - 2], and the quotient is what your teacher wrote. -Jonathan King

score 3 · Accepted Answer · edited Apr 13 '17 at 12:19

3

math.stackexchange.com told me as a way to solve my problem. Here's the solution:

https://math.stackexchange.com/questions/33946/problem-to-determine-the-chromatic-polynomial-of-a-graph

edited Apr 13 '17 at 12:19

Community

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answered Apr 20 '11 at 21:29

franvergara66

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score 0 · Answer 3 · answered Apr 07 '17 at 16:13

In the book I am following (Graph Theory with Applications - Deo Prentice Hall) it is done differently. Instead of excluding the edge they connect two non-adjacent vertices.

Using this technique I am getting

P (G, λ) = 2λ(λ-1)^2(λ-2) + 2λ(λ-1)(λ-2)(λ-3) + λ(λ-1)(λ-2)(λ-3)(λ-4) which is also not equal to either of your results.

problem to determine the chromatic polynomial of a graph

3 Answers3