To prove: The isomorphic graphs have the same chromatic number and the chromatic polynomial.
Explanation of Solution
Definition used:
Chromatic number:
Let
Chromatic polynomial:
For each nonnegative integer k, the number of k-colorings of the vertices of a graph G is denoted by
Description:
Suppose that,
Let
By the isomorphism defined, a vertex u in
That is, the vertex v in
Then,
Similarly, the coloring in
If
Thus, the isomorphism condition implies that the colorings onto
That is, the operations are inverse of each other and thus obtaining a bijection of coloring of the graphs
By the above mentioned definitions, the chromatic number and chromatic colorings depends only on the number of colorings. Here, both the graphs have same colorings.
Therefore, the isomorphic graphs have the same chromatic number and the chromatic polynomial.
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Chapter 12 Solutions
Introductory Combinatorics
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