Question

The following statements are about the chromatic number ?(G) and the chromatic index ?'(G) of graphs. We use ?(G) for the maximum degree of G. Are the following statements true or false? 1. For all n ? 3, ?'(C_n) = ?(C_n). 2. If a graph contains a vertex of degree 3, then the chromatic index of the graph is 3. 3. In any bipartite graph G, ?'(G) = 2. 4. If a graph contains a vertex of degree 3, then the chromatic index of the graph is at least 3.

          The following statements are about the chromatic number ?(G) and the chromatic index ?'(G) of graphs. We use ?(G) for the maximum degree of G. Are the following statements true or false?
1. For all n ? 3, ?'(C_n) = ?(C_n).
2. If a graph contains a vertex of degree 3, then the chromatic index of the graph is 3.
3. In any bipartite graph G, ?'(G) = 2.
4. If a graph contains a vertex of degree 3, then the chromatic index of the graph is at least 3.

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The following statements are about the chromatic number ?(G) and the chromatic index ?'(G) of graphs. We use ?(G) for the maximum degree of G. Are the following statements true or false?
1. For all n ? 3, ?'(Cn) = ?(Cn).
2. If a graph contains a vertex of degree 3, then the chromatic index of the graph is 3.
3. In any bipartite graph G, ?'(G) = 2.
4. If a graph contains a vertex of degree 3, then the chromatic index of the graph is at least 3.

Added by Melissa S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/10/2023

Step 1

The chromatic index of a cycle graph $C_n$ is given by $X(C_n) = 3$ if $n$ is odd and $X(C_n) = 2$ if $n$ is even. So, the statement "For all $n > 3$, $X(C_n) = 4(C_n)$" is false. Show more…

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The following statements are about the chromatic number χ(G) and the chromatic index χ'(G) of graphs. We use Δ(G) for the maximum degree of G. Are the following statements true or false? 1. For all n ≥ 3, χ'(C_n) = Δ(C_n). 2. If a graph contains a vertex of degree 3, then the chromatic index of the graph is 3. 3. In any bipartite graph G, χ'(G) = 2. 4. If a graph contains a vertex of degree 3, then the chromatic index of the graph is at least 3.

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