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Chapter 11: Problem 5

a) Determine \(P(G, \lambda)\) for \(G=K_{1,3}\). b) For \(n \in \mathbf{Z}^{+}\), what is the chromatic polynomial for \(K_{1, n}\) ? What is its chromatic number?

Short Answer

Expert verified
a) For \(G=K_{1,3}\), the chromatic polynomial \(P(G, \lambda)=\lambda(\lambda-1)^3\). b) For any \(n \in \mathbf{Z}^{+}\), the chromatic polynomial for \(K_{1, n}\) is \(P(G, \lambda)=\lambda(\lambda-1)^n\) and its chromatic number is 2.

Step by step solution

01

Determine \(P(G, \lambda)\) for \(G=K_{1,3}\)

A \(K_{1,3}\) graph is a star graph with 1 center and 3 leaves. Since the leaves can be colored with any of \(\lambda-1\) colors that are not the same as the color of the center, and the center can be colored with any of \(\lambda\) colors, the chromatic polynomial is \(P(G, \lambda)=\lambda(\lambda-1)^3\).
02

Determine \(P(G, \lambda)\) for \(G=K_{1, n}\)

Applying the same reasoning as in Step 1 to a star graph with 1 center and \(n\) leaves, one finds that the center can be colored with any of \(\lambda\) colors and the \(n\) leaves can each be colored with any of \(\lambda-1\) colors that are not the same as the color of the center. This gives a chromatic polynomial of \(P(G, \lambda)=\lambda(\lambda-1)^n\).
03

Determine the chromatic number for \(G=K_{1, n}\)

The chromatic number is the smallest number of colors such that adjacent vertices do not have the same color. For a star graph, since only one vertex connects with all others, the chromatic number is 2 - one color for the central node and another for the rest.

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Most popular questions from this chapter

Prove that any subgraph of a bipartite graph is bipartite.

a) How many subgraphs \(H=(V, E)\) of \(K_{6}\) satisfy \(|V|=3 ?\) (If two subgraphs are isomorphic but have different vertex sets, consider them distinct.) b) How many subgraphs \(H=(V, E)\) of \(K_{6}\) satisfy \(|V|=4 ?\) c) How many subgraphs does \(K_{6}\) have?

For \(n \in \mathrm{Z}^{+}\)where \(n \geq 4\), let \(V^{\prime}=\left\\{v_{1}, v_{2}, v_{3}, \ldots, v_{n-1}\right\\}\) be the vertex set for the complete graph \(K_{n-1}\). Construct the loop-free undirected graph \(H_{n}=(V, E)\) from \(K_{n-1}\) as follows: \(V=V^{\prime} \cup\\{v\\}\), and \(E\) consists of all the edges in \(K_{n-1}\) together with the new edge \(\left\\{v, v_{1}\right\\}\). a) Show that \(H_{n}\) has a Hamilton path but no Hamilton cycle. b) How large is the edge set \(E\) ?

Let \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\). How many subgraphs of \(K_{n}\) are isomorphic to the complete bipartite graph \(K_{1,3}\) ?

If \(n \geq 3\), how many different Hamilton cycles are there in the wheel graph \(W_{n} ?\)
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