5 colors available  how many proper colorings  are possible to color the grpah  K3,3

To solve this problem, we need to determine how many proper colorings are possible for the graph \( K_{3,3} \) using 5 available colors.

### Step 1: Understanding the Graph
The graph \( K_{3,3} \) is a **complete bipartite graph** with 3 vertices in each partition. A proper coloring of a graph means no two adjacent vertices share the same color.

- \( K_{3,3} \) consists of two disjoint sets of vertices, say \( \{v_1, v_2, v_3\} \) and \( \{u_1, u_2, u_3\} \), where every vertex in the first set is connected to every vertex in the second set.
- There are no edges between vertices within the same set.

### Step 2: Bipartite Graph Coloring
In a bipartite graph, the vertices within one partition can all be the same color, as there are no edges between them. However, the two partitions must have different colors to maintain proper coloring.

- The 3 vertices in set \( \{v_1, v_2, v_3\} \) can all be colored the same, and similarly for the 3 vertices in set \( \{u_1, u_2, u_3\} \).
- The two sets \( \{v_1, v_2, v_3\} \) and \( \{u_1, u_2, u_3\} \) must have different colors.

### Step 3: Counting the Colorings
1. For the first set \( \{v_1, v_2, v_3\} \), you can choose any of the 5 colors.
2. For the second set \( \{u_1, u_2, u_3\} \), you can choose any color except the color chosen for the first set. This gives 4 possible choices.

### Step 4: Total Number of Proper Colorings
Thus, the total number of proper colorings is:

\[
5 \times 4 = 20
\]

### Conclusion:
There are **20 proper colorings** possible for the graph \( K_{3,3} \) with 5 available colors.

Would you like a further breakdown or more details on graph coloring?

Here are 5 related questions:
1. How would the result change if there were 6 colors available?
2. What is the chromatic number of \( K_{3,3} \)?
3. How do bipartite graphs differ from complete graphs in terms of coloring?
4. What is the number of proper colorings for a general \( K_{m,n} \) bipartite graph?
5. How does the number of proper colorings change for a non-bipartite graph?

**Tip:** In bipartite graphs, the chromatic number is always 2, which simplifies proper colorings.

5 colors available, how many proper colorings are possible to color the graph K3,3?

Discover how to determine the number of proper colorings for the bipartite graph K3,3 using 5 colors. This problem explores graph theory concepts such as proper coloring and bipartite graphs, providing a detailed solution for high school students in Grades 10-12.

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