The problem you have shared involves finding the chromatic number of each of the five given graphs. The chromatic number is the minimum number of colors required to color the vertices of a graph such that no two adjacent vertices share the same color.

Let's analyze each graph one by one:

1. First Graph (Cycle with 6 Vertices)

• This graph is a simple cycle with 6 vertices (an even number).
• A cycle with an even number of vertices requires 2 colors to color it since you can alternate colors between adjacent vertices.

Minimum number of colors: 2

2. Second Graph (Complete Graph with 5 Vertices)

• This graph is a complete graph, $K_5$, where every vertex is connected to every other vertex.
• In a complete graph with $n$ vertices, the chromatic number is $n$ because each vertex must have a distinct color (since each pair of vertices is adjacent).

Minimum number of colors: 5

3. Third Graph (Grid-like Graph)

• This graph is a bipartite grid with 8 vertices arranged in two rows of 4.
• Bipartite graphs can be colored with just 2 colors, since the vertices can be divided into two sets where no vertices within the same set are adjacent.

Minimum number of colors: 2

4. Fourth Graph (Complete Bipartite Graph)

• This graph is a complete bipartite graph, $K_{3,3}$, with two sets of 3 vertices where each vertex in one set is connected to all vertices in the other set.
• Complete bipartite graphs are always 2-colorable because each set of vertices can be colored with one of the two colors.

Minimum number of colors: 2

5. Fifth Graph (Pentagon with Internal Connections)

• This graph is a pentagon (cycle of 5 vertices) with an internal vertex connected to all other vertices, creating a wheel graph $W_5$.
• A wheel graph with an odd number of outer vertices (in this case, 5) requires 3 colors. The central vertex requires its own color, and the outer cycle can be colored with two alternating colors.

Minimum number of colors: 3

Summary of Chromatic Numbers:

1. Cycle with 6 vertices: 2 colors
2. Complete graph $K_5$: 5 colors
3. Grid-like graph: 2 colors
4. Complete bipartite graph $K_{3,3}$: 2 colors
5. Wheel graph $W_5$: 3 colors

These are the minimum numbers of colors needed for each graph, as justified by their respective structures.

Let me know if you would like further details or have any questions!

Follow-up questions:

1. How does the chromatic number change for an odd cycle graph?
2. What is the chromatic number of a complete bipartite graph $K_{m,n}$?
3. Can you explain why a complete graph always requires $n$ colors where $n$ is the number of vertices?
4. How does the chromatic number of a wheel graph change as the number of outer vertices increases?
5. What is the chromatic number for a grid graph with different dimensions?

Tip:

The chromatic number of a graph gives a good understanding of how "dense" the graph is in terms of vertex connections!