Question

Problem 12.29. A basic example of a simple graph with chromatic number $n$ is the complete graph on $n$ vertices, that is $chi(K_n) = n$. This implies that any graph with $K_n$ as a subgraph must have chromatic number at least $n$. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that contains no triangle---length three cycle---and hence no subgraph isomorphic to $K_n$ for $n ge 3$. Namely, let $G$ be the 11-vertex graph of Figure 12.27. The reader can verify that $G$ is triangle-free. Figure 12.27 Graph $G$ with no triangles and $chi(G) = 4$. (a) Show that $G$ is 4-colorable. (b) Prove that $G$ can't be colored with 3 colors.

          Problem 12.29.
A basic example of a simple graph with chromatic number $n$ is the complete graph
on $n$ vertices, that is $chi(K_n) = n$. This implies that any graph with $K_n$ as a subgraph
must have chromatic number at least $n$. It's a common misconception to think that,
conversely, graphs with high chromatic number must contain a large complete sub-
graph. In this problem we exhibit a simple example countering this misconception,
namely a graph with chromatic number four that contains no triangle---length three
cycle---and hence no subgraph isomorphic to $K_n$ for $n ge 3$. Namely, let $G$ be the
11-vertex graph of Figure 12.27. The reader can verify that $G$ is triangle-free.
Figure 12.27 Graph $G$ with no triangles and $chi(G) = 4$.
(a) Show that $G$ is 4-colorable.
(b) Prove that $G$ can't be colored with 3 colors.

Problem 12.29.
A basic example of a simple graph with chromatic number n is the complete graph
on n vertices, that is chi(Kn) = n. This implies that any graph with Kn as a subgraph
must have chromatic number at least n. It's a common misconception to think that,
conversely, graphs with high chromatic number must contain a large complete sub-
graph. In this problem we exhibit a simple example countering this misconception,
namely a graph with chromatic number four that contains no triangle—length three
cycle—and hence no subgraph isomorphic to Kn for n ge 3. Namely, let G be the
11-vertex graph of Figure 12.27. The reader can verify that G is triangle-free.
Figure 12.27 Graph G with no triangles and chi(G) = 4.
(a) Show that G is 4-colorable.
(b) Prove that G can't be colored with 3 colors.

Added by Betty E.

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