Question 28

The graph contains 6 vertices with 4 outer vertices and 2 inner vertices. Examining the subgraphs reveals triangles (3-cliques), such as the cycle formed by an outer vertex and its adjacent inner and outer neighbors. The presence of a $K_3$ implies that the chromatic number $\chi(G) \ge 3$.

Let us check whether a 3-coloring is possible using colors $\{1, 2, 3\}$:

• Assign color 1 to vertex $v_1$ and vertex $u_2$.
• Assign color 2 to vertex $v_2$ and vertex $v_3$.
• Assign color 3 to vertex $u_1$ and vertex $v_4$.

Checking all edges confirms that no two adjacent vertices share the same color. Since the graph cannot be colored with 2 colors (it contains odd cycles) but can be colored with 3 colors, its chromatic number is 3.