Question

Being provided with the graph below: (i) Determine the chromatic polynomial alongside the chromatic number of the given graph. (ii) Provide a minimal proper coloring of that graph Additional Guideline: This question can be tackled using the known properties of chromatic polynomials. Can you regularly color it with 0, 1, 2 colors? Probably not, so it is p(G, ?) = ?(? - 1)(? - 2)(?² + a? + b) (5 nodes meaning degree 5, leading coefficient 1). Moreover, it must start like ?? - 8?? + ? (8 edges), what can you conclude concerning the value of a? Have you tried to color it with 3 colors? Then, you must have found there are 6 ways to do that, and p(G,3)=6 gives you the value of b.

          Being provided with the graph below:
(i) Determine the chromatic polynomial alongside the chromatic number of the given graph.
(ii) Provide a minimal proper coloring of that graph
Additional Guideline:
This question can be tackled using the known properties of chromatic polynomials.
Can you regularly color it with 0, 1, 2 colors?
Probably not, so it is p(G, ?) = ?(? - 1)(? - 2)(?² + a? + b) (5 nodes meaning degree 5, leading coefficient 1).
Moreover, it must start like ?? - 8?? + ? (8 edges), what can you conclude concerning the value of a?
Have you tried to color it with 3 colors?
Then, you must have found there are 6 ways to do that, and p(G,3)=6 gives you the value of b.

Being provided with the graph below:
(i) Determine the chromatic polynomial alongside the chromatic number of the given graph.
(ii) Provide a minimal proper coloring of that graph
Additional Guideline:
This question can be tackled using the known properties of chromatic polynomials.
Can you regularly color it with 0, 1, 2 colors?
Probably not, so it is p(G, ?) = ?(? - 1)(? - 2)(?² + a? + b) (5 nodes meaning degree 5, leading coefficient 1).
Moreover, it must start like ?? - 8?? + ? (8 edges), what can you conclude concerning the value of a?
Have you tried to color it with 3 colors?
Then, you must have found there are 6 ways to do that, and p(G,3)=6 gives you the value of b.

Added by Mike N.

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