Prove that isomorphic graphs have the same chromatic number and the same chromatic polynomial.
I'm having a difficult time with this proof, and I don't know where to start. If you can, can you please explain how to go about the proof?
Prove that isomorphic graphs have the same chromatic number and the same chromatic polynomial.
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combinatorics
graph-theory
coloring
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0Suppose otherwise. Without loss of generality, let the two graphs be labeled $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ with the chromatic number of $G_2$ strictly higher than that of $G_1$. But then as they are isomorphic there is a relabeling of the edges and vertices of $G_1$ that transforms $G_1$ into $G_2$. Composing this with the coloring we get a coloring of $G_2$ such that [insert more details here and reach a conclusion]. – 2017-02-23
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0Please use the body of the Question to pose explicitly the problem you want help to solve. Putting the problem statement only in the title, as you've done here, invites confusion as Readers guess as what your real difficulty or interest is. The body of the Question is intended for a full statement of problems and the associated context. – 2017-10-13
1 Answers
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It's quite simply a corrollary of the following observation:
Suppose $G_1 =(V_1 ,E_1)$ and $G_2 = (V_2, E_2)$ are two graphs and $f: V_1 \rightarrow V_2$ is a graph isomporphism between them (so a bijection of vertices such that $(v, w) \in E_1$ iff $(f(v), f(w)) \in E_2$).
If now $c: V_1 \rightarrow \underline{n} = \{1,\ldots,n\}$ is a vertex-colouring of $G_1$with $n$ colours then check that $c \circ f^{-1}: V_2 \rightarrow \underline{n}$ is a vertex colouring of $G_2$. We can also transform a colouring $c'$ on $G_2$ to one on $G_1$ via $f$ as well: use $c' \circ f$. The isomorphism condition ensures that valid colourings go to valid colourings (with the same number of colours). Check that these operations are each other's inverse, so we have a bijection of colourings (of $G_1$ and $G_2$) with a given number of colours.
As the chromatic number/polynomial only depends on the existence or number of colourings with a certain number of colours, these must be the same for isomorphic graphs.