if 5 colors are avialable how many proper colorings re possibler to color graph k3,3

To calculate how many proper colorings are possible for the graph \( K_{3,3} \) with 5 available colors, we need to understand the properties of the graph and the rules for proper coloring.

1. **Graph \( K_{3,3} \)**:
   - \( K_{3,3} \) is a complete bipartite graph. It has two sets of 3 vertices each, where every vertex in one set is connected to every vertex in the other set.
   - There are no edges between vertices within the same set.

2. **Proper coloring**:
   - A proper coloring means no two adjacent vertices can share the same color.

### Step-by-step calculation:
- Each set of \( K_{3,3} \) has 3 vertices.
- Since there are no edges within the same set, all 3 vertices in one set can be assigned the same color.
- However, each vertex in one set must have a different color from each vertex in the other set because they are all adjacent to each other.

### For each set:
- Let’s color the first set of 3 vertices. We can choose any of the 5 colors for the first vertex.
  - For the second set, all vertices must be colored with a different color from those in the first set, leaving us 2 colors to choose from.

Thus, the number of proper colorings would be:

\[
5 \times 5 \times 5 = 

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

Discover how to calculate the number of proper colorings for the bipartite graph K3,3 with 5 available colors. This problem introduces key graph theory concepts like bipartite graphs and proper coloring, suitable for undergraduate-level students.

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