The chromatic number of the following graph is ______.

To determine the chromatic number $\chi(G)$ (the minimum number of colors needed to color the vertices such that no two adjacent vertices share the same color):

1. Lower Bound: The graph contains triangles (cliques of size 3), for example the subgraph induced by vertices $\{a, b, c\}$. Hence, at least 3 colors are required:

   $$\chi(G) \ge 3$$

2. 3-Coloring Assignment: Let the three colors be $1, 2, 3$:

   • Assign color $1$ to vertex $a$: $c(a) = 1$

   • Assign color $2$ to vertex $b$: $c(b) = 2$

   • Assign color $3$ to vertex $c$: $c(c) = 3$

   • Vertex $e$ is adjacent to $a (1)$ and $c (3)$, so assign $c(e) = 2$.

   • Vertex $d$ is adjacent to $c (3)$ and $e (2)$, so assign $c(d) = 1$.

   • Vertex $f$ is adjacent to $b (2)$, $e (2)$, and $d (1)$, so assign $c(f) = 3$.

3. Verification: All adjacent vertex pairs have distinct colors:

   • Edges with color pairs: $(a, b): 1-2$, $(a, c): 1-3$, $(a, e): 1-2$, $(b, c): 2-3$, $(b, f): 2-3$, $(c, d): 3-1$, $(c, e): 3-2$, $(d, e): 1-2$, $(d, f): 1-3$, $(e, f): 2-3$.

Since a valid 3-coloring exists and the graph cannot be colored with 2 colors, the chromatic number is 3.